Properties

Label 28-15e28-1.1-c3e14-0-0
Degree $28$
Conductor $8.522\times 10^{32}$
Sign $1$
Analytic cond. $5.28056\times 10^{15}$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 5·3-s + 12·4-s − 10·6-s + 22·7-s − 20·8-s + 21·9-s + 23·11-s + 60·12-s + 96·13-s − 44·14-s + 59·16-s + 322·17-s − 42·18-s + 558·19-s + 110·21-s − 46·22-s − 96·23-s − 100·24-s − 192·26-s + 220·27-s + 264·28-s − 296·29-s − 244·31-s + 12·32-s + 115·33-s − 644·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.962·3-s + 3/2·4-s − 0.680·6-s + 1.18·7-s − 0.883·8-s + 7/9·9-s + 0.630·11-s + 1.44·12-s + 2.04·13-s − 0.839·14-s + 0.921·16-s + 4.59·17-s − 0.549·18-s + 6.73·19-s + 1.14·21-s − 0.445·22-s − 0.870·23-s − 0.850·24-s − 1.44·26-s + 1.56·27-s + 1.78·28-s − 1.89·29-s − 1.41·31-s + 0.0662·32-s + 0.606·33-s − 3.24·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{28}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(28\)
Conductor: \(3^{28} \cdot 5^{28}\)
Sign: $1$
Analytic conductor: \(5.28056\times 10^{15}\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((28,\ 3^{28} \cdot 5^{28} ,\ ( \ : [3/2]^{14} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(287.3415974\)
\(L(\frac12)\) \(\approx\) \(287.3415974\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5 T + 4 T^{2} - 5 p^{3} T^{3} + 427 p T^{4} - 362 p^{2} T^{5} + 829 p^{3} T^{6} - 13 p^{9} T^{7} + 829 p^{6} T^{8} - 362 p^{8} T^{9} + 427 p^{10} T^{10} - 5 p^{15} T^{11} + 4 p^{15} T^{12} - 5 p^{18} T^{13} + p^{21} T^{14} \)
5 \( 1 \)
good2 \( 1 + p T - p^{3} T^{2} - 5 p^{2} T^{3} + 37 T^{4} + 3 p^{3} T^{5} - 23 p^{3} T^{6} + 173 p T^{7} - 871 T^{8} - 3179 p^{2} T^{9} - 927 p^{3} T^{10} + 2803 p^{5} T^{11} + 1861 p^{7} T^{12} - 299 p^{9} T^{13} - 3759 p^{9} T^{14} - 299 p^{12} T^{15} + 1861 p^{13} T^{16} + 2803 p^{14} T^{17} - 927 p^{15} T^{18} - 3179 p^{17} T^{19} - 871 p^{18} T^{20} + 173 p^{22} T^{21} - 23 p^{27} T^{22} + 3 p^{30} T^{23} + 37 p^{30} T^{24} - 5 p^{35} T^{25} - p^{39} T^{26} + p^{40} T^{27} + p^{42} T^{28} \)
7 \( 1 - 22 T - 346 T^{2} + 20008 T^{3} - 158749 T^{4} - 704510 p T^{5} + 170302844 T^{6} - 1241272856 T^{7} - 54533077780 T^{8} + 1561453660484 T^{9} + 824567922400 T^{10} - 574414217025882 T^{11} + 7092925398656976 T^{12} + 11454966041061366 p T^{13} - 67080217878248868 p^{2} T^{14} + 11454966041061366 p^{4} T^{15} + 7092925398656976 p^{6} T^{16} - 574414217025882 p^{9} T^{17} + 824567922400 p^{12} T^{18} + 1561453660484 p^{15} T^{19} - 54533077780 p^{18} T^{20} - 1241272856 p^{21} T^{21} + 170302844 p^{24} T^{22} - 704510 p^{28} T^{23} - 158749 p^{30} T^{24} + 20008 p^{33} T^{25} - 346 p^{36} T^{26} - 22 p^{39} T^{27} + p^{42} T^{28} \)
11 \( 1 - 23 T - 4622 T^{2} + 156515 T^{3} + 749552 p T^{4} - 373295985 T^{5} - 838645094 p T^{6} + 367854523937 T^{7} + 18612254564513 T^{8} - 16613032308302 p T^{9} - 43965548274791280 T^{10} + 330029679280370662 T^{11} + 66707501960674286009 T^{12} - \)\(33\!\cdots\!25\)\( T^{13} - \)\(83\!\cdots\!86\)\( T^{14} - \)\(33\!\cdots\!25\)\( p^{3} T^{15} + 66707501960674286009 p^{6} T^{16} + 330029679280370662 p^{9} T^{17} - 43965548274791280 p^{12} T^{18} - 16613032308302 p^{16} T^{19} + 18612254564513 p^{18} T^{20} + 367854523937 p^{21} T^{21} - 838645094 p^{25} T^{22} - 373295985 p^{27} T^{23} + 749552 p^{31} T^{24} + 156515 p^{33} T^{25} - 4622 p^{36} T^{26} - 23 p^{39} T^{27} + p^{42} T^{28} \)
13 \( 1 - 96 T - 2975 T^{2} + 487064 T^{3} + 1125472 T^{4} - 728668420 T^{5} - 38153654081 T^{6} + 1311488532476 T^{7} + 216607089579541 T^{8} - 6307824711072176 T^{9} - 515041558744564490 T^{10} + 11288979491598794464 T^{11} + \)\(11\!\cdots\!41\)\( T^{12} - \)\(54\!\cdots\!44\)\( T^{13} - \)\(28\!\cdots\!21\)\( T^{14} - \)\(54\!\cdots\!44\)\( p^{3} T^{15} + \)\(11\!\cdots\!41\)\( p^{6} T^{16} + 11288979491598794464 p^{9} T^{17} - 515041558744564490 p^{12} T^{18} - 6307824711072176 p^{15} T^{19} + 216607089579541 p^{18} T^{20} + 1311488532476 p^{21} T^{21} - 38153654081 p^{24} T^{22} - 728668420 p^{27} T^{23} + 1125472 p^{30} T^{24} + 487064 p^{33} T^{25} - 2975 p^{36} T^{26} - 96 p^{39} T^{27} + p^{42} T^{28} \)
17 \( ( 1 - 161 T + 24747 T^{2} - 2620018 T^{3} + 242733069 T^{4} - 19012626063 T^{5} + 1428967389431 T^{6} - 5791825964988 p T^{7} + 1428967389431 p^{3} T^{8} - 19012626063 p^{6} T^{9} + 242733069 p^{9} T^{10} - 2620018 p^{12} T^{11} + 24747 p^{15} T^{12} - 161 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
19 \( ( 1 - 279 T + 63455 T^{2} - 9627646 T^{3} + 1292727231 T^{4} - 139417672577 T^{5} + 13934197302465 T^{6} - 1188087181318116 T^{7} + 13934197302465 p^{3} T^{8} - 139417672577 p^{6} T^{9} + 1292727231 p^{9} T^{10} - 9627646 p^{12} T^{11} + 63455 p^{15} T^{12} - 279 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
23 \( 1 + 96 T - 61514 T^{2} - 3466752 T^{3} + 2279148427 T^{4} + 2758807746 p T^{5} - 59876297207752 T^{6} - 815406725889402 T^{7} + 1212095044479634376 T^{8} + 7570252354791898158 T^{9} - \)\(20\!\cdots\!28\)\( T^{10} - \)\(67\!\cdots\!54\)\( T^{11} + \)\(30\!\cdots\!48\)\( T^{12} + \)\(16\!\cdots\!36\)\( p T^{13} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(16\!\cdots\!36\)\( p^{4} T^{15} + \)\(30\!\cdots\!48\)\( p^{6} T^{16} - \)\(67\!\cdots\!54\)\( p^{9} T^{17} - \)\(20\!\cdots\!28\)\( p^{12} T^{18} + 7570252354791898158 p^{15} T^{19} + 1212095044479634376 p^{18} T^{20} - 815406725889402 p^{21} T^{21} - 59876297207752 p^{24} T^{22} + 2758807746 p^{28} T^{23} + 2279148427 p^{30} T^{24} - 3466752 p^{33} T^{25} - 61514 p^{36} T^{26} + 96 p^{39} T^{27} + p^{42} T^{28} \)
29 \( 1 + 296 T - 66806 T^{2} - 13330316 T^{3} + 6358604533 T^{4} + 580937886288 T^{5} - 274778709332800 T^{6} + 2073504722795524 T^{7} + 9929070933174490424 T^{8} - \)\(48\!\cdots\!12\)\( T^{9} - \)\(18\!\cdots\!96\)\( T^{10} + \)\(22\!\cdots\!12\)\( T^{11} + \)\(23\!\cdots\!42\)\( T^{12} - \)\(20\!\cdots\!68\)\( T^{13} - \)\(81\!\cdots\!56\)\( T^{14} - \)\(20\!\cdots\!68\)\( p^{3} T^{15} + \)\(23\!\cdots\!42\)\( p^{6} T^{16} + \)\(22\!\cdots\!12\)\( p^{9} T^{17} - \)\(18\!\cdots\!96\)\( p^{12} T^{18} - \)\(48\!\cdots\!12\)\( p^{15} T^{19} + 9929070933174490424 p^{18} T^{20} + 2073504722795524 p^{21} T^{21} - 274778709332800 p^{24} T^{22} + 580937886288 p^{27} T^{23} + 6358604533 p^{30} T^{24} - 13330316 p^{33} T^{25} - 66806 p^{36} T^{26} + 296 p^{39} T^{27} + p^{42} T^{28} \)
31 \( 1 + 244 T - 79417 T^{2} - 9259252 T^{3} + 7221915764 T^{4} + 314634162392 T^{5} - 277630233328651 T^{6} + 25254086777734424 T^{7} + 8436197722541960369 T^{8} - \)\(14\!\cdots\!28\)\( T^{9} + \)\(79\!\cdots\!14\)\( T^{10} + \)\(64\!\cdots\!84\)\( T^{11} - \)\(10\!\cdots\!23\)\( T^{12} - \)\(56\!\cdots\!88\)\( T^{13} + \)\(52\!\cdots\!69\)\( T^{14} - \)\(56\!\cdots\!88\)\( p^{3} T^{15} - \)\(10\!\cdots\!23\)\( p^{6} T^{16} + \)\(64\!\cdots\!84\)\( p^{9} T^{17} + \)\(79\!\cdots\!14\)\( p^{12} T^{18} - \)\(14\!\cdots\!28\)\( p^{15} T^{19} + 8436197722541960369 p^{18} T^{20} + 25254086777734424 p^{21} T^{21} - 277630233328651 p^{24} T^{22} + 314634162392 p^{27} T^{23} + 7221915764 p^{30} T^{24} - 9259252 p^{33} T^{25} - 79417 p^{36} T^{26} + 244 p^{39} T^{27} + p^{42} T^{28} \)
37 \( ( 1 + 404 T + 7243 p T^{2} + 88433668 T^{3} + 32537859593 T^{4} + 8785146905212 T^{5} + 2408172804310695 T^{6} + 541287630181037208 T^{7} + 2408172804310695 p^{3} T^{8} + 8785146905212 p^{6} T^{9} + 32537859593 p^{9} T^{10} + 88433668 p^{12} T^{11} + 7243 p^{16} T^{12} + 404 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
41 \( 1 + 47 T - 318887 T^{2} + 39133822 T^{3} + 1426731800 p T^{4} - 12736813723812 T^{5} - 6014913089457667 T^{6} + 2129991519337143073 T^{7} + \)\(35\!\cdots\!04\)\( T^{8} - \)\(21\!\cdots\!85\)\( T^{9} - \)\(19\!\cdots\!17\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{11} - \)\(16\!\cdots\!65\)\( T^{12} - \)\(37\!\cdots\!09\)\( T^{13} + \)\(16\!\cdots\!94\)\( T^{14} - \)\(37\!\cdots\!09\)\( p^{3} T^{15} - \)\(16\!\cdots\!65\)\( p^{6} T^{16} + \)\(13\!\cdots\!88\)\( p^{9} T^{17} - \)\(19\!\cdots\!17\)\( p^{12} T^{18} - \)\(21\!\cdots\!85\)\( p^{15} T^{19} + \)\(35\!\cdots\!04\)\( p^{18} T^{20} + 2129991519337143073 p^{21} T^{21} - 6014913089457667 p^{24} T^{22} - 12736813723812 p^{27} T^{23} + 1426731800 p^{31} T^{24} + 39133822 p^{33} T^{25} - 318887 p^{36} T^{26} + 47 p^{39} T^{27} + p^{42} T^{28} \)
43 \( 1 - 525 T - 186026 T^{2} + 111741005 T^{3} + 28616000908 T^{4} - 13570705475683 T^{5} - 3776510352412802 T^{6} + 1143764976924110675 T^{7} + \)\(40\!\cdots\!65\)\( T^{8} - \)\(62\!\cdots\!86\)\( T^{9} - \)\(39\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!70\)\( T^{11} + \)\(36\!\cdots\!93\)\( T^{12} - \)\(49\!\cdots\!43\)\( T^{13} - \)\(30\!\cdots\!54\)\( T^{14} - \)\(49\!\cdots\!43\)\( p^{3} T^{15} + \)\(36\!\cdots\!93\)\( p^{6} T^{16} + \)\(21\!\cdots\!70\)\( p^{9} T^{17} - \)\(39\!\cdots\!72\)\( p^{12} T^{18} - \)\(62\!\cdots\!86\)\( p^{15} T^{19} + \)\(40\!\cdots\!65\)\( p^{18} T^{20} + 1143764976924110675 p^{21} T^{21} - 3776510352412802 p^{24} T^{22} - 13570705475683 p^{27} T^{23} + 28616000908 p^{30} T^{24} + 111741005 p^{33} T^{25} - 186026 p^{36} T^{26} - 525 p^{39} T^{27} + p^{42} T^{28} \)
47 \( 1 + 164 T - 432074 T^{2} - 145513532 T^{3} + 84379305835 T^{4} + 40904764422702 T^{5} - 7514604533217820 T^{6} - 5543939360309950310 T^{7} + \)\(33\!\cdots\!28\)\( T^{8} + \)\(27\!\cdots\!90\)\( T^{9} - \)\(91\!\cdots\!40\)\( T^{10} + \)\(16\!\cdots\!34\)\( T^{11} + \)\(29\!\cdots\!44\)\( T^{12} - \)\(18\!\cdots\!00\)\( T^{13} - \)\(42\!\cdots\!52\)\( T^{14} - \)\(18\!\cdots\!00\)\( p^{3} T^{15} + \)\(29\!\cdots\!44\)\( p^{6} T^{16} + \)\(16\!\cdots\!34\)\( p^{9} T^{17} - \)\(91\!\cdots\!40\)\( p^{12} T^{18} + \)\(27\!\cdots\!90\)\( p^{15} T^{19} + \)\(33\!\cdots\!28\)\( p^{18} T^{20} - 5543939360309950310 p^{21} T^{21} - 7514604533217820 p^{24} T^{22} + 40904764422702 p^{27} T^{23} + 84379305835 p^{30} T^{24} - 145513532 p^{33} T^{25} - 432074 p^{36} T^{26} + 164 p^{39} T^{27} + p^{42} T^{28} \)
53 \( ( 1 - 506 T + 541623 T^{2} - 172442740 T^{3} + 132333548265 T^{4} - 35763427746966 T^{5} + 25492045997195975 T^{6} - 6232790423469661848 T^{7} + 25492045997195975 p^{3} T^{8} - 35763427746966 p^{6} T^{9} + 132333548265 p^{9} T^{10} - 172442740 p^{12} T^{11} + 541623 p^{15} T^{12} - 506 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
59 \( 1 + 85 T - 324116 T^{2} - 37822603 T^{3} - 18072986498 T^{4} - 749566914441 T^{5} + 18104485680180662 T^{6} + 2325621683130615665 T^{7} - \)\(23\!\cdots\!89\)\( T^{8} - \)\(37\!\cdots\!76\)\( T^{9} - \)\(76\!\cdots\!32\)\( T^{10} - \)\(14\!\cdots\!28\)\( T^{11} + \)\(24\!\cdots\!07\)\( T^{12} + \)\(75\!\cdots\!91\)\( T^{13} + \)\(34\!\cdots\!30\)\( T^{14} + \)\(75\!\cdots\!91\)\( p^{3} T^{15} + \)\(24\!\cdots\!07\)\( p^{6} T^{16} - \)\(14\!\cdots\!28\)\( p^{9} T^{17} - \)\(76\!\cdots\!32\)\( p^{12} T^{18} - \)\(37\!\cdots\!76\)\( p^{15} T^{19} - \)\(23\!\cdots\!89\)\( p^{18} T^{20} + 2325621683130615665 p^{21} T^{21} + 18104485680180662 p^{24} T^{22} - 749566914441 p^{27} T^{23} - 18072986498 p^{30} T^{24} - 37822603 p^{33} T^{25} - 324116 p^{36} T^{26} + 85 p^{39} T^{27} + p^{42} T^{28} \)
61 \( 1 + 828 T - 452210 T^{2} - 369231764 T^{3} + 278788731013 T^{4} + 167143149611332 T^{5} - 56118284670137180 T^{6} - 18642400601799152372 T^{7} + \)\(12\!\cdots\!36\)\( T^{8} + \)\(19\!\cdots\!40\)\( T^{9} + \)\(31\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{11} - \)\(12\!\cdots\!50\)\( T^{12} - \)\(69\!\cdots\!68\)\( T^{13} + \)\(52\!\cdots\!88\)\( T^{14} - \)\(69\!\cdots\!68\)\( p^{3} T^{15} - \)\(12\!\cdots\!50\)\( p^{6} T^{16} + \)\(14\!\cdots\!28\)\( p^{9} T^{17} + \)\(31\!\cdots\!52\)\( p^{12} T^{18} + \)\(19\!\cdots\!40\)\( p^{15} T^{19} + \)\(12\!\cdots\!36\)\( p^{18} T^{20} - 18642400601799152372 p^{21} T^{21} - 56118284670137180 p^{24} T^{22} + 167143149611332 p^{27} T^{23} + 278788731013 p^{30} T^{24} - 369231764 p^{33} T^{25} - 452210 p^{36} T^{26} + 828 p^{39} T^{27} + p^{42} T^{28} \)
67 \( 1 - 1093 T - 822457 T^{2} + 1193636152 T^{3} + 454158371864 T^{4} - 751860756542282 T^{5} - 203216506929239641 T^{6} + \)\(31\!\cdots\!03\)\( T^{7} + \)\(94\!\cdots\!56\)\( T^{8} - \)\(99\!\cdots\!45\)\( T^{9} - \)\(44\!\cdots\!95\)\( T^{10} + \)\(21\!\cdots\!94\)\( T^{11} + \)\(18\!\cdots\!67\)\( T^{12} - \)\(23\!\cdots\!99\)\( T^{13} - \)\(61\!\cdots\!10\)\( T^{14} - \)\(23\!\cdots\!99\)\( p^{3} T^{15} + \)\(18\!\cdots\!67\)\( p^{6} T^{16} + \)\(21\!\cdots\!94\)\( p^{9} T^{17} - \)\(44\!\cdots\!95\)\( p^{12} T^{18} - \)\(99\!\cdots\!45\)\( p^{15} T^{19} + \)\(94\!\cdots\!56\)\( p^{18} T^{20} + \)\(31\!\cdots\!03\)\( p^{21} T^{21} - 203216506929239641 p^{24} T^{22} - 751860756542282 p^{27} T^{23} + 454158371864 p^{30} T^{24} + 1193636152 p^{33} T^{25} - 822457 p^{36} T^{26} - 1093 p^{39} T^{27} + p^{42} T^{28} \)
71 \( ( 1 + 328 T + 1276365 T^{2} + 268163756 T^{3} + 813660594393 T^{4} + 105690507940488 T^{5} + 374340014427629213 T^{6} + 36698946645864837192 T^{7} + 374340014427629213 p^{3} T^{8} + 105690507940488 p^{6} T^{9} + 813660594393 p^{9} T^{10} + 268163756 p^{12} T^{11} + 1276365 p^{15} T^{12} + 328 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
73 \( ( 1 + 2085 T + 4050563 T^{2} + 4984423486 T^{3} + 5631570806793 T^{4} + 4895773027898939 T^{5} + 3922468709989734339 T^{6} + \)\(25\!\cdots\!16\)\( T^{7} + 3922468709989734339 p^{3} T^{8} + 4895773027898939 p^{6} T^{9} + 5631570806793 p^{9} T^{10} + 4984423486 p^{12} T^{11} + 4050563 p^{15} T^{12} + 2085 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
79 \( 1 + 2110 T + 985703 T^{2} - 835227010 T^{3} - 781830943216 T^{4} + 464399626886 p T^{5} + 2897229142548143 p T^{6} + \)\(17\!\cdots\!78\)\( T^{7} + \)\(61\!\cdots\!29\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{9} + \)\(31\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!52\)\( T^{11} + \)\(51\!\cdots\!61\)\( T^{12} - \)\(30\!\cdots\!18\)\( T^{13} - \)\(40\!\cdots\!59\)\( T^{14} - \)\(30\!\cdots\!18\)\( p^{3} T^{15} + \)\(51\!\cdots\!61\)\( p^{6} T^{16} + \)\(12\!\cdots\!52\)\( p^{9} T^{17} + \)\(31\!\cdots\!42\)\( p^{12} T^{18} - \)\(58\!\cdots\!80\)\( p^{15} T^{19} + \)\(61\!\cdots\!29\)\( p^{18} T^{20} + \)\(17\!\cdots\!78\)\( p^{21} T^{21} + 2897229142548143 p^{25} T^{22} + 464399626886 p^{28} T^{23} - 781830943216 p^{30} T^{24} - 835227010 p^{33} T^{25} + 985703 p^{36} T^{26} + 2110 p^{39} T^{27} + p^{42} T^{28} \)
83 \( 1 + 1290 T - 1618274 T^{2} - 2147915856 T^{3} + 2057042647747 T^{4} + 2283305981146542 T^{5} - 1857167019426755668 T^{6} - \)\(18\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!24\)\( T^{8} + \)\(10\!\cdots\!20\)\( T^{9} - \)\(51\!\cdots\!92\)\( T^{10} - \)\(45\!\cdots\!14\)\( T^{11} + \)\(18\!\cdots\!16\)\( T^{12} + \)\(10\!\cdots\!78\)\( T^{13} - \)\(64\!\cdots\!32\)\( T^{14} + \)\(10\!\cdots\!78\)\( p^{3} T^{15} + \)\(18\!\cdots\!16\)\( p^{6} T^{16} - \)\(45\!\cdots\!14\)\( p^{9} T^{17} - \)\(51\!\cdots\!92\)\( p^{12} T^{18} + \)\(10\!\cdots\!20\)\( p^{15} T^{19} + \)\(11\!\cdots\!24\)\( p^{18} T^{20} - \)\(18\!\cdots\!20\)\( p^{21} T^{21} - 1857167019426755668 p^{24} T^{22} + 2283305981146542 p^{27} T^{23} + 2057042647747 p^{30} T^{24} - 2147915856 p^{33} T^{25} - 1618274 p^{36} T^{26} + 1290 p^{39} T^{27} + p^{42} T^{28} \)
89 \( ( 1 - 3048 T + 5723102 T^{2} - 6323602086 T^{3} + 4534976524791 T^{4} - 780789931274556 T^{5} - 1899913405596725465 T^{6} + \)\(27\!\cdots\!50\)\( T^{7} - 1899913405596725465 p^{3} T^{8} - 780789931274556 p^{6} T^{9} + 4534976524791 p^{9} T^{10} - 6323602086 p^{12} T^{11} + 5723102 p^{15} T^{12} - 3048 p^{18} T^{13} + p^{21} T^{14} )^{2} \)
97 \( 1 - 1787 T - 890874 T^{2} + 2279797707 T^{3} + 825970464582 T^{4} - 599429743810785 T^{5} - 2342205601130520192 T^{6} + \)\(53\!\cdots\!47\)\( T^{7} + \)\(19\!\cdots\!35\)\( T^{8} - \)\(11\!\cdots\!84\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(91\!\cdots\!24\)\( T^{11} - \)\(13\!\cdots\!73\)\( T^{12} + \)\(11\!\cdots\!87\)\( T^{13} + \)\(20\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!87\)\( p^{3} T^{15} - \)\(13\!\cdots\!73\)\( p^{6} T^{16} - \)\(91\!\cdots\!24\)\( p^{9} T^{17} + \)\(11\!\cdots\!04\)\( p^{12} T^{18} - \)\(11\!\cdots\!84\)\( p^{15} T^{19} + \)\(19\!\cdots\!35\)\( p^{18} T^{20} + \)\(53\!\cdots\!47\)\( p^{21} T^{21} - 2342205601130520192 p^{24} T^{22} - 599429743810785 p^{27} T^{23} + 825970464582 p^{30} T^{24} + 2279797707 p^{33} T^{25} - 890874 p^{36} T^{26} - 1787 p^{39} T^{27} + p^{42} T^{28} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.24160094636352588120137883253, −3.18938249886562503496248048635, −3.18114114526405575889635024724, −3.16157489309598086665766024539, −3.02345488713653315440347069493, −2.71060414474042798112181051567, −2.66579942806813224169788850124, −2.50070813824748647952774964365, −2.33353000074340086590932724641, −2.29756460036312101105234260547, −2.08310862588058315058340387983, −1.92708502888368088453311177365, −1.68584119660571791847269750351, −1.61293851890339520914912192910, −1.55333102184869040779465892175, −1.54214320749757017199127086241, −1.51868884519345636625275681799, −1.13492680310427990804844668461, −1.06855555372178669050909311743, −1.04937979076290453807642734220, −1.01990580137945005037246974423, −0.67028637445166750810915321413, −0.62795536976154033942543766677, −0.39084935801532509033003841444, −0.37168556973712492837050373942, 0.37168556973712492837050373942, 0.39084935801532509033003841444, 0.62795536976154033942543766677, 0.67028637445166750810915321413, 1.01990580137945005037246974423, 1.04937979076290453807642734220, 1.06855555372178669050909311743, 1.13492680310427990804844668461, 1.51868884519345636625275681799, 1.54214320749757017199127086241, 1.55333102184869040779465892175, 1.61293851890339520914912192910, 1.68584119660571791847269750351, 1.92708502888368088453311177365, 2.08310862588058315058340387983, 2.29756460036312101105234260547, 2.33353000074340086590932724641, 2.50070813824748647952774964365, 2.66579942806813224169788850124, 2.71060414474042798112181051567, 3.02345488713653315440347069493, 3.16157489309598086665766024539, 3.18114114526405575889635024724, 3.18938249886562503496248048635, 3.24160094636352588120137883253

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

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