Properties

Label 24-91e12-1.1-c9e12-0-0
Degree $24$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $1.12343\times 10^{20}$
Root an. cond. $6.84603$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 21·2-s − 323·3-s − 1.37e3·4-s − 5.20e3·5-s + 6.78e3·6-s + 2.88e4·7-s + 2.71e4·8-s − 2.66e4·9-s + 1.09e5·10-s − 8.00e4·11-s + 4.45e5·12-s + 3.42e5·13-s − 6.05e5·14-s + 1.68e6·15-s + 7.56e5·16-s − 1.49e6·17-s + 5.60e5·18-s − 1.09e5·19-s + 7.17e6·20-s − 9.30e6·21-s + 1.68e6·22-s − 3.36e6·23-s − 8.77e6·24-s + 1.78e6·25-s − 7.19e6·26-s + 1.92e7·27-s − 3.97e7·28-s + ⋯
L(s)  = 1  − 0.928·2-s − 2.30·3-s − 2.69·4-s − 3.72·5-s + 2.13·6-s + 4.53·7-s + 2.34·8-s − 1.35·9-s + 3.45·10-s − 1.64·11-s + 6.20·12-s + 3.32·13-s − 4.20·14-s + 8.56·15-s + 2.88·16-s − 4.33·17-s + 1.25·18-s − 0.191·19-s + 10.0·20-s − 10.4·21-s + 1.53·22-s − 2.50·23-s − 5.39·24-s + 0.914·25-s − 3.08·26-s + 6.97·27-s − 12.2·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(1.12343\times 10^{20}\)
Root analytic conductor: \(6.84603\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 7^{12} \cdot 13^{12} ,\ ( \ : [9/2]^{12} ),\ 1 )\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 - p^{4} T )^{12} \)
13 \( ( 1 - p^{4} T )^{12} \)
good2 \( 1 + 21 T + 455 p^{2} T^{2} + 20013 p T^{3} + 505833 p^{2} T^{4} + 1271301 p^{5} T^{5} + 52997873 p^{5} T^{6} + 28396137 p^{10} T^{7} + 4334131989 p^{8} T^{8} + 8198375019 p^{11} T^{9} + 37955122897 p^{14} T^{10} + 266646642819 p^{15} T^{11} + 4943298411869 p^{16} T^{12} + 266646642819 p^{24} T^{13} + 37955122897 p^{32} T^{14} + 8198375019 p^{38} T^{15} + 4334131989 p^{44} T^{16} + 28396137 p^{55} T^{17} + 52997873 p^{59} T^{18} + 1271301 p^{68} T^{19} + 505833 p^{74} T^{20} + 20013 p^{82} T^{21} + 455 p^{92} T^{22} + 21 p^{99} T^{23} + p^{108} T^{24} \)
3 \( 1 + 323 T + 131003 T^{2} + 3520184 p^{2} T^{3} + 8035548808 T^{4} + 1612479734992 T^{5} + 328005956025323 T^{6} + 19311015742089995 p T^{7} + 1143022620063611575 p^{2} T^{8} + 20308451973842943706 p^{4} T^{9} + \)\(32\!\cdots\!06\)\( p^{4} T^{10} + \)\(15\!\cdots\!40\)\( p^{5} T^{11} + \)\(77\!\cdots\!80\)\( p^{6} T^{12} + \)\(15\!\cdots\!40\)\( p^{14} T^{13} + \)\(32\!\cdots\!06\)\( p^{22} T^{14} + 20308451973842943706 p^{31} T^{15} + 1143022620063611575 p^{38} T^{16} + 19311015742089995 p^{46} T^{17} + 328005956025323 p^{54} T^{18} + 1612479734992 p^{63} T^{19} + 8035548808 p^{72} T^{20} + 3520184 p^{83} T^{21} + 131003 p^{90} T^{22} + 323 p^{99} T^{23} + p^{108} T^{24} \)
5 \( 1 + 5202 T + 25274892 T^{2} + 16593748026 p T^{3} + 49773305914176 p T^{4} + 616850062541595714 T^{5} + \)\(14\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!58\)\( T^{7} + \)\(10\!\cdots\!04\)\( p T^{8} + \)\(15\!\cdots\!42\)\( p^{4} T^{9} + \)\(12\!\cdots\!88\)\( p^{3} T^{10} + \)\(37\!\cdots\!94\)\( p^{4} T^{11} + \)\(22\!\cdots\!34\)\( p^{6} T^{12} + \)\(37\!\cdots\!94\)\( p^{13} T^{13} + \)\(12\!\cdots\!88\)\( p^{21} T^{14} + \)\(15\!\cdots\!42\)\( p^{31} T^{15} + \)\(10\!\cdots\!04\)\( p^{37} T^{16} + \)\(28\!\cdots\!58\)\( p^{45} T^{17} + \)\(14\!\cdots\!28\)\( p^{54} T^{18} + 616850062541595714 p^{63} T^{19} + 49773305914176 p^{73} T^{20} + 16593748026 p^{82} T^{21} + 25274892 p^{90} T^{22} + 5202 p^{99} T^{23} + p^{108} T^{24} \)
11 \( 1 + 80061 T + 14611322095 T^{2} + 781712783774892 T^{3} + 7607965485132603040 p T^{4} + \)\(29\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!07\)\( T^{6} + \)\(49\!\cdots\!47\)\( T^{7} + \)\(53\!\cdots\!67\)\( T^{8} + \)\(17\!\cdots\!42\)\( T^{9} + \)\(99\!\cdots\!58\)\( T^{10} - \)\(86\!\cdots\!12\)\( T^{11} + \)\(21\!\cdots\!24\)\( T^{12} - \)\(86\!\cdots\!12\)\( p^{9} T^{13} + \)\(99\!\cdots\!58\)\( p^{18} T^{14} + \)\(17\!\cdots\!42\)\( p^{27} T^{15} + \)\(53\!\cdots\!67\)\( p^{36} T^{16} + \)\(49\!\cdots\!47\)\( p^{45} T^{17} + \)\(25\!\cdots\!07\)\( p^{54} T^{18} + \)\(29\!\cdots\!92\)\( p^{63} T^{19} + 7607965485132603040 p^{73} T^{20} + 781712783774892 p^{81} T^{21} + 14611322095 p^{90} T^{22} + 80061 p^{99} T^{23} + p^{108} T^{24} \)
17 \( 1 + 1493598 T + 1682013694772 T^{2} + 1289559694683879642 T^{3} + \)\(83\!\cdots\!10\)\( T^{4} + \)\(43\!\cdots\!50\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} + \)\(83\!\cdots\!02\)\( T^{7} + \)\(32\!\cdots\!83\)\( T^{8} + \)\(11\!\cdots\!40\)\( T^{9} + \)\(43\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!96\)\( T^{11} + \)\(54\!\cdots\!64\)\( T^{12} + \)\(15\!\cdots\!96\)\( p^{9} T^{13} + \)\(43\!\cdots\!60\)\( p^{18} T^{14} + \)\(11\!\cdots\!40\)\( p^{27} T^{15} + \)\(32\!\cdots\!83\)\( p^{36} T^{16} + \)\(83\!\cdots\!02\)\( p^{45} T^{17} + \)\(20\!\cdots\!20\)\( p^{54} T^{18} + \)\(43\!\cdots\!50\)\( p^{63} T^{19} + \)\(83\!\cdots\!10\)\( p^{72} T^{20} + 1289559694683879642 p^{81} T^{21} + 1682013694772 p^{90} T^{22} + 1493598 p^{99} T^{23} + p^{108} T^{24} \)
19 \( 1 + 109038 T + 1875118477120 T^{2} + 367411858677613098 T^{3} + \)\(16\!\cdots\!68\)\( T^{4} + \)\(42\!\cdots\!58\)\( T^{5} + \)\(98\!\cdots\!76\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{7} + \)\(43\!\cdots\!72\)\( T^{8} + \)\(10\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!96\)\( T^{10} + \)\(31\!\cdots\!70\)\( T^{11} + \)\(53\!\cdots\!54\)\( T^{12} + \)\(31\!\cdots\!70\)\( p^{9} T^{13} + \)\(16\!\cdots\!96\)\( p^{18} T^{14} + \)\(10\!\cdots\!86\)\( p^{27} T^{15} + \)\(43\!\cdots\!72\)\( p^{36} T^{16} + \)\(25\!\cdots\!70\)\( p^{45} T^{17} + \)\(98\!\cdots\!76\)\( p^{54} T^{18} + \)\(42\!\cdots\!58\)\( p^{63} T^{19} + \)\(16\!\cdots\!68\)\( p^{72} T^{20} + 367411858677613098 p^{81} T^{21} + 1875118477120 p^{90} T^{22} + 109038 p^{99} T^{23} + p^{108} T^{24} \)
23 \( 1 + 3367443 T + 13631130500551 T^{2} + 32858906345761716624 T^{3} + \)\(87\!\cdots\!90\)\( T^{4} + \)\(17\!\cdots\!70\)\( T^{5} + \)\(37\!\cdots\!21\)\( T^{6} + \)\(66\!\cdots\!73\)\( T^{7} + \)\(12\!\cdots\!84\)\( T^{8} + \)\(19\!\cdots\!21\)\( T^{9} + \)\(30\!\cdots\!11\)\( T^{10} + \)\(42\!\cdots\!78\)\( T^{11} + \)\(61\!\cdots\!56\)\( T^{12} + \)\(42\!\cdots\!78\)\( p^{9} T^{13} + \)\(30\!\cdots\!11\)\( p^{18} T^{14} + \)\(19\!\cdots\!21\)\( p^{27} T^{15} + \)\(12\!\cdots\!84\)\( p^{36} T^{16} + \)\(66\!\cdots\!73\)\( p^{45} T^{17} + \)\(37\!\cdots\!21\)\( p^{54} T^{18} + \)\(17\!\cdots\!70\)\( p^{63} T^{19} + \)\(87\!\cdots\!90\)\( p^{72} T^{20} + 32858906345761716624 p^{81} T^{21} + 13631130500551 p^{90} T^{22} + 3367443 p^{99} T^{23} + p^{108} T^{24} \)
29 \( 1 + 459762 p T + 203464152865072 T^{2} + \)\(18\!\cdots\!06\)\( T^{3} + \)\(17\!\cdots\!00\)\( T^{4} + \)\(11\!\cdots\!94\)\( T^{5} + \)\(82\!\cdots\!80\)\( T^{6} + \)\(47\!\cdots\!38\)\( T^{7} + \)\(26\!\cdots\!40\)\( T^{8} + \)\(13\!\cdots\!46\)\( T^{9} + \)\(61\!\cdots\!56\)\( T^{10} + \)\(25\!\cdots\!22\)\( T^{11} + \)\(10\!\cdots\!46\)\( T^{12} + \)\(25\!\cdots\!22\)\( p^{9} T^{13} + \)\(61\!\cdots\!56\)\( p^{18} T^{14} + \)\(13\!\cdots\!46\)\( p^{27} T^{15} + \)\(26\!\cdots\!40\)\( p^{36} T^{16} + \)\(47\!\cdots\!38\)\( p^{45} T^{17} + \)\(82\!\cdots\!80\)\( p^{54} T^{18} + \)\(11\!\cdots\!94\)\( p^{63} T^{19} + \)\(17\!\cdots\!00\)\( p^{72} T^{20} + \)\(18\!\cdots\!06\)\( p^{81} T^{21} + 203464152865072 p^{90} T^{22} + 459762 p^{100} T^{23} + p^{108} T^{24} \)
31 \( 1 + 3954765 T + 5304196903125 p T^{2} + \)\(42\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} + \)\(15\!\cdots\!38\)\( T^{5} + \)\(62\!\cdots\!93\)\( T^{6} - \)\(20\!\cdots\!89\)\( T^{7} + \)\(21\!\cdots\!04\)\( T^{8} - \)\(46\!\cdots\!05\)\( T^{9} + \)\(58\!\cdots\!87\)\( T^{10} - \)\(22\!\cdots\!34\)\( T^{11} + \)\(15\!\cdots\!84\)\( T^{12} - \)\(22\!\cdots\!34\)\( p^{9} T^{13} + \)\(58\!\cdots\!87\)\( p^{18} T^{14} - \)\(46\!\cdots\!05\)\( p^{27} T^{15} + \)\(21\!\cdots\!04\)\( p^{36} T^{16} - \)\(20\!\cdots\!89\)\( p^{45} T^{17} + \)\(62\!\cdots\!93\)\( p^{54} T^{18} + \)\(15\!\cdots\!38\)\( p^{63} T^{19} + \)\(12\!\cdots\!58\)\( p^{72} T^{20} + \)\(42\!\cdots\!60\)\( p^{81} T^{21} + 5304196903125 p^{91} T^{22} + 3954765 p^{99} T^{23} + p^{108} T^{24} \)
37 \( 1 - 580535 T + 682861143456725 T^{2} - \)\(10\!\cdots\!30\)\( T^{3} + \)\(22\!\cdots\!30\)\( T^{4} + \)\(34\!\cdots\!18\)\( T^{5} + \)\(44\!\cdots\!49\)\( T^{6} + \)\(22\!\cdots\!17\)\( T^{7} + \)\(59\!\cdots\!31\)\( T^{8} + \)\(68\!\cdots\!86\)\( T^{9} + \)\(57\!\cdots\!42\)\( T^{10} + \)\(13\!\cdots\!16\)\( T^{11} + \)\(58\!\cdots\!96\)\( T^{12} + \)\(13\!\cdots\!16\)\( p^{9} T^{13} + \)\(57\!\cdots\!42\)\( p^{18} T^{14} + \)\(68\!\cdots\!86\)\( p^{27} T^{15} + \)\(59\!\cdots\!31\)\( p^{36} T^{16} + \)\(22\!\cdots\!17\)\( p^{45} T^{17} + \)\(44\!\cdots\!49\)\( p^{54} T^{18} + \)\(34\!\cdots\!18\)\( p^{63} T^{19} + \)\(22\!\cdots\!30\)\( p^{72} T^{20} - \)\(10\!\cdots\!30\)\( p^{81} T^{21} + 682861143456725 p^{90} T^{22} - 580535 p^{99} T^{23} + p^{108} T^{24} \)
41 \( 1 + 27018171 T + 1795159595410169 T^{2} + \)\(41\!\cdots\!66\)\( T^{3} + \)\(15\!\cdots\!82\)\( T^{4} + \)\(32\!\cdots\!10\)\( T^{5} + \)\(95\!\cdots\!61\)\( T^{6} + \)\(18\!\cdots\!71\)\( T^{7} + \)\(46\!\cdots\!07\)\( T^{8} + \)\(84\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!94\)\( T^{10} + \)\(32\!\cdots\!12\)\( T^{11} + \)\(67\!\cdots\!48\)\( T^{12} + \)\(32\!\cdots\!12\)\( p^{9} T^{13} + \)\(19\!\cdots\!94\)\( p^{18} T^{14} + \)\(84\!\cdots\!06\)\( p^{27} T^{15} + \)\(46\!\cdots\!07\)\( p^{36} T^{16} + \)\(18\!\cdots\!71\)\( p^{45} T^{17} + \)\(95\!\cdots\!61\)\( p^{54} T^{18} + \)\(32\!\cdots\!10\)\( p^{63} T^{19} + \)\(15\!\cdots\!82\)\( p^{72} T^{20} + \)\(41\!\cdots\!66\)\( p^{81} T^{21} + 1795159595410169 p^{90} T^{22} + 27018171 p^{99} T^{23} + p^{108} T^{24} \)
43 \( 1 - 31237588 T + 2257776903762696 T^{2} - \)\(31\!\cdots\!68\)\( T^{3} + \)\(21\!\cdots\!84\)\( T^{4} - \)\(21\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!64\)\( T^{6} - \)\(16\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!68\)\( T^{8} - \)\(56\!\cdots\!88\)\( T^{9} + \)\(80\!\cdots\!68\)\( T^{10} - \)\(34\!\cdots\!60\)\( T^{11} + \)\(46\!\cdots\!58\)\( T^{12} - \)\(34\!\cdots\!60\)\( p^{9} T^{13} + \)\(80\!\cdots\!68\)\( p^{18} T^{14} - \)\(56\!\cdots\!88\)\( p^{27} T^{15} + \)\(13\!\cdots\!68\)\( p^{36} T^{16} - \)\(16\!\cdots\!12\)\( p^{45} T^{17} + \)\(19\!\cdots\!64\)\( p^{54} T^{18} - \)\(21\!\cdots\!12\)\( p^{63} T^{19} + \)\(21\!\cdots\!84\)\( p^{72} T^{20} - \)\(31\!\cdots\!68\)\( p^{81} T^{21} + 2257776903762696 p^{90} T^{22} - 31237588 p^{99} T^{23} + p^{108} T^{24} \)
47 \( 1 + 21983709 T + 6449736326551947 T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!02\)\( T^{4} + \)\(49\!\cdots\!74\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} + \)\(97\!\cdots\!87\)\( T^{7} + \)\(63\!\cdots\!72\)\( T^{8} + \)\(13\!\cdots\!51\)\( T^{9} + \)\(82\!\cdots\!99\)\( T^{10} + \)\(16\!\cdots\!98\)\( T^{11} + \)\(95\!\cdots\!20\)\( T^{12} + \)\(16\!\cdots\!98\)\( p^{9} T^{13} + \)\(82\!\cdots\!99\)\( p^{18} T^{14} + \)\(13\!\cdots\!51\)\( p^{27} T^{15} + \)\(63\!\cdots\!72\)\( p^{36} T^{16} + \)\(97\!\cdots\!87\)\( p^{45} T^{17} + \)\(40\!\cdots\!57\)\( p^{54} T^{18} + \)\(49\!\cdots\!74\)\( p^{63} T^{19} + \)\(19\!\cdots\!02\)\( p^{72} T^{20} + \)\(15\!\cdots\!28\)\( p^{81} T^{21} + 6449736326551947 p^{90} T^{22} + 21983709 p^{99} T^{23} + p^{108} T^{24} \)
53 \( 1 + 196548234 T + 35198517058763368 T^{2} + \)\(43\!\cdots\!86\)\( T^{3} + \)\(48\!\cdots\!72\)\( T^{4} + \)\(46\!\cdots\!78\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} + \)\(30\!\cdots\!98\)\( T^{7} + \)\(21\!\cdots\!56\)\( T^{8} + \)\(14\!\cdots\!14\)\( T^{9} + \)\(91\!\cdots\!92\)\( T^{10} + \)\(55\!\cdots\!46\)\( T^{11} + \)\(32\!\cdots\!50\)\( T^{12} + \)\(55\!\cdots\!46\)\( p^{9} T^{13} + \)\(91\!\cdots\!92\)\( p^{18} T^{14} + \)\(14\!\cdots\!14\)\( p^{27} T^{15} + \)\(21\!\cdots\!56\)\( p^{36} T^{16} + \)\(30\!\cdots\!98\)\( p^{45} T^{17} + \)\(39\!\cdots\!40\)\( p^{54} T^{18} + \)\(46\!\cdots\!78\)\( p^{63} T^{19} + \)\(48\!\cdots\!72\)\( p^{72} T^{20} + \)\(43\!\cdots\!86\)\( p^{81} T^{21} + 35198517058763368 p^{90} T^{22} + 196548234 p^{99} T^{23} + p^{108} T^{24} \)
59 \( 1 + 215907906 T + 93051297603453312 T^{2} + \)\(15\!\cdots\!82\)\( T^{3} + \)\(38\!\cdots\!66\)\( T^{4} + \)\(52\!\cdots\!14\)\( T^{5} + \)\(93\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!34\)\( T^{7} + \)\(15\!\cdots\!47\)\( T^{8} + \)\(15\!\cdots\!76\)\( T^{9} + \)\(19\!\cdots\!56\)\( T^{10} + \)\(17\!\cdots\!60\)\( T^{11} + \)\(18\!\cdots\!12\)\( T^{12} + \)\(17\!\cdots\!60\)\( p^{9} T^{13} + \)\(19\!\cdots\!56\)\( p^{18} T^{14} + \)\(15\!\cdots\!76\)\( p^{27} T^{15} + \)\(15\!\cdots\!47\)\( p^{36} T^{16} + \)\(10\!\cdots\!34\)\( p^{45} T^{17} + \)\(93\!\cdots\!52\)\( p^{54} T^{18} + \)\(52\!\cdots\!14\)\( p^{63} T^{19} + \)\(38\!\cdots\!66\)\( p^{72} T^{20} + \)\(15\!\cdots\!82\)\( p^{81} T^{21} + 93051297603453312 p^{90} T^{22} + 215907906 p^{99} T^{23} + p^{108} T^{24} \)
61 \( 1 + 218340705 T + 100131243662012405 T^{2} + \)\(18\!\cdots\!42\)\( T^{3} + \)\(47\!\cdots\!94\)\( T^{4} + \)\(76\!\cdots\!26\)\( T^{5} + \)\(14\!\cdots\!77\)\( T^{6} + \)\(20\!\cdots\!65\)\( T^{7} + \)\(32\!\cdots\!71\)\( T^{8} + \)\(40\!\cdots\!10\)\( T^{9} + \)\(54\!\cdots\!66\)\( T^{10} + \)\(61\!\cdots\!24\)\( T^{11} + \)\(72\!\cdots\!04\)\( T^{12} + \)\(61\!\cdots\!24\)\( p^{9} T^{13} + \)\(54\!\cdots\!66\)\( p^{18} T^{14} + \)\(40\!\cdots\!10\)\( p^{27} T^{15} + \)\(32\!\cdots\!71\)\( p^{36} T^{16} + \)\(20\!\cdots\!65\)\( p^{45} T^{17} + \)\(14\!\cdots\!77\)\( p^{54} T^{18} + \)\(76\!\cdots\!26\)\( p^{63} T^{19} + \)\(47\!\cdots\!94\)\( p^{72} T^{20} + \)\(18\!\cdots\!42\)\( p^{81} T^{21} + 100131243662012405 p^{90} T^{22} + 218340705 p^{99} T^{23} + p^{108} T^{24} \)
67 \( 1 - 14544775 T + 141694138994957475 T^{2} - \)\(75\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!40\)\( T^{4} - \)\(82\!\cdots\!08\)\( T^{5} + \)\(61\!\cdots\!43\)\( T^{6} - \)\(52\!\cdots\!41\)\( T^{7} + \)\(26\!\cdots\!51\)\( T^{8} - \)\(23\!\cdots\!86\)\( T^{9} + \)\(95\!\cdots\!38\)\( T^{10} - \)\(78\!\cdots\!12\)\( T^{11} + \)\(28\!\cdots\!44\)\( T^{12} - \)\(78\!\cdots\!12\)\( p^{9} T^{13} + \)\(95\!\cdots\!38\)\( p^{18} T^{14} - \)\(23\!\cdots\!86\)\( p^{27} T^{15} + \)\(26\!\cdots\!51\)\( p^{36} T^{16} - \)\(52\!\cdots\!41\)\( p^{45} T^{17} + \)\(61\!\cdots\!43\)\( p^{54} T^{18} - \)\(82\!\cdots\!08\)\( p^{63} T^{19} + \)\(11\!\cdots\!40\)\( p^{72} T^{20} - \)\(75\!\cdots\!60\)\( p^{81} T^{21} + 141694138994957475 p^{90} T^{22} - 14544775 p^{99} T^{23} + p^{108} T^{24} \)
71 \( 1 + 552451776 T + 414690808109535172 T^{2} + \)\(17\!\cdots\!20\)\( T^{3} + \)\(77\!\cdots\!66\)\( T^{4} + \)\(26\!\cdots\!88\)\( T^{5} + \)\(90\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{7} + \)\(74\!\cdots\!51\)\( T^{8} + \)\(18\!\cdots\!64\)\( T^{9} + \)\(47\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(23\!\cdots\!84\)\( T^{12} + \)\(10\!\cdots\!88\)\( p^{9} T^{13} + \)\(47\!\cdots\!92\)\( p^{18} T^{14} + \)\(18\!\cdots\!64\)\( p^{27} T^{15} + \)\(74\!\cdots\!51\)\( p^{36} T^{16} + \)\(25\!\cdots\!80\)\( p^{45} T^{17} + \)\(90\!\cdots\!72\)\( p^{54} T^{18} + \)\(26\!\cdots\!88\)\( p^{63} T^{19} + \)\(77\!\cdots\!66\)\( p^{72} T^{20} + \)\(17\!\cdots\!20\)\( p^{81} T^{21} + 414690808109535172 p^{90} T^{22} + 552451776 p^{99} T^{23} + p^{108} T^{24} \)
73 \( 1 + 349395159 T + 433258671860550013 T^{2} + \)\(13\!\cdots\!62\)\( T^{3} + \)\(94\!\cdots\!64\)\( T^{4} + \)\(25\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} + \)\(33\!\cdots\!19\)\( T^{7} + \)\(14\!\cdots\!00\)\( T^{8} + \)\(32\!\cdots\!61\)\( T^{9} + \)\(12\!\cdots\!57\)\( T^{10} + \)\(24\!\cdots\!68\)\( T^{11} + \)\(81\!\cdots\!10\)\( T^{12} + \)\(24\!\cdots\!68\)\( p^{9} T^{13} + \)\(12\!\cdots\!57\)\( p^{18} T^{14} + \)\(32\!\cdots\!61\)\( p^{27} T^{15} + \)\(14\!\cdots\!00\)\( p^{36} T^{16} + \)\(33\!\cdots\!19\)\( p^{45} T^{17} + \)\(13\!\cdots\!71\)\( p^{54} T^{18} + \)\(25\!\cdots\!28\)\( p^{63} T^{19} + \)\(94\!\cdots\!64\)\( p^{72} T^{20} + \)\(13\!\cdots\!62\)\( p^{81} T^{21} + 433258671860550013 p^{90} T^{22} + 349395159 p^{99} T^{23} + p^{108} T^{24} \)
79 \( 1 - 962249727 T + 1329356175636328815 T^{2} - \)\(90\!\cdots\!72\)\( T^{3} + \)\(75\!\cdots\!98\)\( T^{4} - \)\(41\!\cdots\!62\)\( T^{5} + \)\(25\!\cdots\!97\)\( T^{6} - \)\(12\!\cdots\!45\)\( T^{7} + \)\(62\!\cdots\!44\)\( T^{8} - \)\(24\!\cdots\!33\)\( T^{9} + \)\(11\!\cdots\!87\)\( T^{10} - \)\(39\!\cdots\!10\)\( T^{11} + \)\(15\!\cdots\!32\)\( T^{12} - \)\(39\!\cdots\!10\)\( p^{9} T^{13} + \)\(11\!\cdots\!87\)\( p^{18} T^{14} - \)\(24\!\cdots\!33\)\( p^{27} T^{15} + \)\(62\!\cdots\!44\)\( p^{36} T^{16} - \)\(12\!\cdots\!45\)\( p^{45} T^{17} + \)\(25\!\cdots\!97\)\( p^{54} T^{18} - \)\(41\!\cdots\!62\)\( p^{63} T^{19} + \)\(75\!\cdots\!98\)\( p^{72} T^{20} - \)\(90\!\cdots\!72\)\( p^{81} T^{21} + 1329356175636328815 p^{90} T^{22} - 962249727 p^{99} T^{23} + p^{108} T^{24} \)
83 \( 1 + 2032575912 T + 3080378365169314036 T^{2} + \)\(31\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!92\)\( T^{4} + \)\(17\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} + \)\(46\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!76\)\( T^{8} + \)\(45\!\cdots\!44\)\( T^{9} + \)\(52\!\cdots\!68\)\( T^{10} - \)\(37\!\cdots\!12\)\( T^{11} - \)\(23\!\cdots\!30\)\( T^{12} - \)\(37\!\cdots\!12\)\( p^{9} T^{13} + \)\(52\!\cdots\!68\)\( p^{18} T^{14} + \)\(45\!\cdots\!44\)\( p^{27} T^{15} + \)\(17\!\cdots\!76\)\( p^{36} T^{16} + \)\(46\!\cdots\!92\)\( p^{45} T^{17} + \)\(10\!\cdots\!96\)\( p^{54} T^{18} + \)\(17\!\cdots\!48\)\( p^{63} T^{19} + \)\(26\!\cdots\!92\)\( p^{72} T^{20} + \)\(31\!\cdots\!60\)\( p^{81} T^{21} + 3080378365169314036 p^{90} T^{22} + 2032575912 p^{99} T^{23} + p^{108} T^{24} \)
89 \( 1 + 280821684 T + 2449495597400795932 T^{2} + \)\(38\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!20\)\( T^{4} + \)\(13\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!92\)\( T^{6} - \)\(78\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!60\)\( T^{8} - \)\(96\!\cdots\!64\)\( T^{9} + \)\(56\!\cdots\!36\)\( T^{10} - \)\(49\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!78\)\( T^{12} - \)\(49\!\cdots\!84\)\( p^{9} T^{13} + \)\(56\!\cdots\!36\)\( p^{18} T^{14} - \)\(96\!\cdots\!64\)\( p^{27} T^{15} + \)\(12\!\cdots\!60\)\( p^{36} T^{16} - \)\(78\!\cdots\!52\)\( p^{45} T^{17} + \)\(21\!\cdots\!92\)\( p^{54} T^{18} + \)\(13\!\cdots\!96\)\( p^{63} T^{19} + \)\(28\!\cdots\!20\)\( p^{72} T^{20} + \)\(38\!\cdots\!20\)\( p^{81} T^{21} + 2449495597400795932 p^{90} T^{22} + 280821684 p^{99} T^{23} + p^{108} T^{24} \)
97 \( 1 + 2115165937 T + 8621574197055286369 T^{2} + \)\(14\!\cdots\!22\)\( T^{3} + \)\(34\!\cdots\!68\)\( T^{4} + \)\(49\!\cdots\!32\)\( T^{5} + \)\(82\!\cdots\!51\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{7} + \)\(13\!\cdots\!20\)\( T^{8} + \)\(14\!\cdots\!55\)\( T^{9} + \)\(16\!\cdots\!05\)\( T^{10} + \)\(14\!\cdots\!60\)\( T^{11} + \)\(14\!\cdots\!78\)\( T^{12} + \)\(14\!\cdots\!60\)\( p^{9} T^{13} + \)\(16\!\cdots\!05\)\( p^{18} T^{14} + \)\(14\!\cdots\!55\)\( p^{27} T^{15} + \)\(13\!\cdots\!20\)\( p^{36} T^{16} + \)\(10\!\cdots\!77\)\( p^{45} T^{17} + \)\(82\!\cdots\!51\)\( p^{54} T^{18} + \)\(49\!\cdots\!32\)\( p^{63} T^{19} + \)\(34\!\cdots\!68\)\( p^{72} T^{20} + \)\(14\!\cdots\!22\)\( p^{81} T^{21} + 8621574197055286369 p^{90} T^{22} + 2115165937 p^{99} T^{23} + p^{108} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.16357146227950475674860914482, −4.00571337574243141469448565024, −3.73262071568285528940593985177, −3.67138169539231515525109397446, −3.57823833065120818841291405005, −3.46504364421280210551555014798, −3.33734785931127690206794973386, −3.23822369745465087511115883025, −3.21733715486111751711618091780, −2.82837065619745434634106761884, −2.59848549064208036613353458827, −2.38465508933771413308684296504, −2.37335264983263087815734409217, −2.36117377561303802496071297074, −2.02451089522675186898418455731, −1.97900869221996322202415656684, −1.86403527468400059545680605573, −1.78559481415174726701306132089, −1.67850962437510708685361609238, −1.48156595976440005263764823808, −1.14749479305637644379452475727, −1.06301772263503479287313519516, −1.04925899222704560976324231974, −1.01677652399347528943642336579, −0.979330092345369840804590889079, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.979330092345369840804590889079, 1.01677652399347528943642336579, 1.04925899222704560976324231974, 1.06301772263503479287313519516, 1.14749479305637644379452475727, 1.48156595976440005263764823808, 1.67850962437510708685361609238, 1.78559481415174726701306132089, 1.86403527468400059545680605573, 1.97900869221996322202415656684, 2.02451089522675186898418455731, 2.36117377561303802496071297074, 2.37335264983263087815734409217, 2.38465508933771413308684296504, 2.59848549064208036613353458827, 2.82837065619745434634106761884, 3.21733715486111751711618091780, 3.23822369745465087511115883025, 3.33734785931127690206794973386, 3.46504364421280210551555014798, 3.57823833065120818841291405005, 3.67138169539231515525109397446, 3.73262071568285528940593985177, 4.00571337574243141469448565024, 4.16357146227950475674860914482

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.