Properties

Label 16-23e8-1.1-c7e8-0-0
Degree $16$
Conductor $78310985281$
Sign $1$
Analytic cond. $7.10140\times 10^{6}$
Root an. cond. $2.68045$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 40·3-s − 192·4-s + 444·5-s + 1.44e3·7-s + 1.05e3·8-s − 1.00e3·9-s + 7.58e3·11-s − 7.68e3·12-s + 1.98e4·13-s + 1.77e4·15-s + 1.40e4·16-s + 4.20e4·17-s + 1.05e3·19-s − 8.52e4·20-s + 5.78e4·21-s − 9.73e4·23-s + 4.23e4·24-s − 1.89e5·25-s − 1.14e5·27-s − 2.77e5·28-s − 1.02e5·29-s + 3.04e5·31-s − 4.07e5·32-s + 3.03e5·33-s + 6.42e5·35-s + 1.93e5·36-s + 2.86e5·37-s + ⋯
L(s)  = 1  + 0.855·3-s − 3/2·4-s + 1.58·5-s + 1.59·7-s + 0.731·8-s − 0.461·9-s + 1.71·11-s − 1.28·12-s + 2.50·13-s + 1.35·15-s + 0.856·16-s + 2.07·17-s + 0.0351·19-s − 2.38·20-s + 1.36·21-s − 1.66·23-s + 0.625·24-s − 2.42·25-s − 1.11·27-s − 2.39·28-s − 0.781·29-s + 1.83·31-s − 2.19·32-s + 1.47·33-s + 2.53·35-s + 0.692·36-s + 0.929·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(23^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(23^{8}\)
Sign: $1$
Analytic conductor: \(7.10140\times 10^{6}\)
Root analytic conductor: \(2.68045\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 23^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(19.77234557\)
\(L(\frac12)\) \(\approx\) \(19.77234557\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( ( 1 + p^{3} T )^{8} \)
good2 \( 1 + 3 p^{6} T^{2} - 1059 T^{3} + 5707 p^{2} T^{4} + 71 p^{3} T^{5} + 219159 p^{4} T^{6} + 853903 p^{4} T^{7} + 5948603 p^{6} T^{8} + 853903 p^{11} T^{9} + 219159 p^{18} T^{10} + 71 p^{24} T^{11} + 5707 p^{30} T^{12} - 1059 p^{35} T^{13} + 3 p^{48} T^{14} + p^{56} T^{16} \)
3 \( 1 - 40 T + 2609 T^{2} - 30700 T^{3} + 6111313 T^{4} - 41955572 p T^{5} + 1720770302 p^{2} T^{6} - 21915987308 p^{3} T^{7} + 631660113670 p^{4} T^{8} - 21915987308 p^{10} T^{9} + 1720770302 p^{16} T^{10} - 41955572 p^{22} T^{11} + 6111313 p^{28} T^{12} - 30700 p^{35} T^{13} + 2609 p^{42} T^{14} - 40 p^{49} T^{15} + p^{56} T^{16} \)
5 \( 1 - 444 T + 77264 p T^{2} - 6007612 p^{2} T^{3} + 577338428 p^{3} T^{4} - 39018985876 p^{4} T^{5} + 2794315915632 p^{5} T^{6} - 164972056629748 p^{6} T^{7} + 9966750053201118 p^{7} T^{8} - 164972056629748 p^{13} T^{9} + 2794315915632 p^{19} T^{10} - 39018985876 p^{25} T^{11} + 577338428 p^{31} T^{12} - 6007612 p^{37} T^{13} + 77264 p^{43} T^{14} - 444 p^{49} T^{15} + p^{56} T^{16} \)
7 \( 1 - 1446 T + 4367332 T^{2} - 5864328662 T^{3} + 9820748098564 T^{4} - 11719145281475054 T^{5} + 13994893447197411676 T^{6} - \)\(14\!\cdots\!90\)\( T^{7} + \)\(13\!\cdots\!02\)\( T^{8} - \)\(14\!\cdots\!90\)\( p^{7} T^{9} + 13994893447197411676 p^{14} T^{10} - 11719145281475054 p^{21} T^{11} + 9820748098564 p^{28} T^{12} - 5864328662 p^{35} T^{13} + 4367332 p^{42} T^{14} - 1446 p^{49} T^{15} + p^{56} T^{16} \)
11 \( 1 - 7588 T + 71548256 T^{2} - 441882720588 T^{3} + 3453974755238028 T^{4} - 17924487921178272444 T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(47\!\cdots\!44\)\( T^{7} + \)\(24\!\cdots\!86\)\( T^{8} - \)\(47\!\cdots\!44\)\( p^{7} T^{9} + \)\(10\!\cdots\!76\)\( p^{14} T^{10} - 17924487921178272444 p^{21} T^{11} + 3453974755238028 p^{28} T^{12} - 441882720588 p^{35} T^{13} + 71548256 p^{42} T^{14} - 7588 p^{49} T^{15} + p^{56} T^{16} \)
13 \( 1 - 19862 T + 422154819 T^{2} - 5862189208474 T^{3} + 75005711331579361 T^{4} - \)\(78\!\cdots\!04\)\( T^{5} + \)\(77\!\cdots\!86\)\( T^{6} - \)\(67\!\cdots\!88\)\( T^{7} + \)\(56\!\cdots\!86\)\( T^{8} - \)\(67\!\cdots\!88\)\( p^{7} T^{9} + \)\(77\!\cdots\!86\)\( p^{14} T^{10} - \)\(78\!\cdots\!04\)\( p^{21} T^{11} + 75005711331579361 p^{28} T^{12} - 5862189208474 p^{35} T^{13} + 422154819 p^{42} T^{14} - 19862 p^{49} T^{15} + p^{56} T^{16} \)
17 \( 1 - 42070 T + 2493711120 T^{2} - 69975983524226 T^{3} + 2562151710632664332 T^{4} - \)\(57\!\cdots\!86\)\( T^{5} + \)\(16\!\cdots\!04\)\( T^{6} - \)\(31\!\cdots\!74\)\( T^{7} + \)\(79\!\cdots\!26\)\( T^{8} - \)\(31\!\cdots\!74\)\( p^{7} T^{9} + \)\(16\!\cdots\!04\)\( p^{14} T^{10} - \)\(57\!\cdots\!86\)\( p^{21} T^{11} + 2562151710632664332 p^{28} T^{12} - 69975983524226 p^{35} T^{13} + 2493711120 p^{42} T^{14} - 42070 p^{49} T^{15} + p^{56} T^{16} \)
19 \( 1 - 1050 T + 2633750012 T^{2} + 30260980504198 T^{3} + 4030939399825765620 T^{4} + \)\(75\!\cdots\!94\)\( T^{5} + \)\(48\!\cdots\!52\)\( T^{6} + \)\(95\!\cdots\!78\)\( T^{7} + \)\(49\!\cdots\!90\)\( T^{8} + \)\(95\!\cdots\!78\)\( p^{7} T^{9} + \)\(48\!\cdots\!52\)\( p^{14} T^{10} + \)\(75\!\cdots\!94\)\( p^{21} T^{11} + 4030939399825765620 p^{28} T^{12} + 30260980504198 p^{35} T^{13} + 2633750012 p^{42} T^{14} - 1050 p^{49} T^{15} + p^{56} T^{16} \)
29 \( 1 + 102578 T + 115287553715 T^{2} + 10213600927463538 T^{3} + \)\(61\!\cdots\!41\)\( T^{4} + \)\(46\!\cdots\!92\)\( T^{5} + \)\(19\!\cdots\!30\)\( T^{6} + \)\(12\!\cdots\!28\)\( T^{7} + \)\(40\!\cdots\!46\)\( T^{8} + \)\(12\!\cdots\!28\)\( p^{7} T^{9} + \)\(19\!\cdots\!30\)\( p^{14} T^{10} + \)\(46\!\cdots\!92\)\( p^{21} T^{11} + \)\(61\!\cdots\!41\)\( p^{28} T^{12} + 10213600927463538 p^{35} T^{13} + 115287553715 p^{42} T^{14} + 102578 p^{49} T^{15} + p^{56} T^{16} \)
31 \( 1 - 9812 p T + 203455697169 T^{2} - 51253279418905452 T^{3} + \)\(18\!\cdots\!89\)\( T^{4} - \)\(38\!\cdots\!48\)\( T^{5} + \)\(98\!\cdots\!86\)\( T^{6} - \)\(16\!\cdots\!48\)\( T^{7} + \)\(33\!\cdots\!50\)\( T^{8} - \)\(16\!\cdots\!48\)\( p^{7} T^{9} + \)\(98\!\cdots\!86\)\( p^{14} T^{10} - \)\(38\!\cdots\!48\)\( p^{21} T^{11} + \)\(18\!\cdots\!89\)\( p^{28} T^{12} - 51253279418905452 p^{35} T^{13} + 203455697169 p^{42} T^{14} - 9812 p^{50} T^{15} + p^{56} T^{16} \)
37 \( 1 - 286472 T + 336267686920 T^{2} - 107758408142193504 T^{3} + \)\(73\!\cdots\!16\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!04\)\( T^{6} - \)\(27\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!30\)\( T^{8} - \)\(27\!\cdots\!36\)\( p^{7} T^{9} + \)\(10\!\cdots\!04\)\( p^{14} T^{10} - \)\(20\!\cdots\!00\)\( p^{21} T^{11} + \)\(73\!\cdots\!16\)\( p^{28} T^{12} - 107758408142193504 p^{35} T^{13} + 336267686920 p^{42} T^{14} - 286472 p^{49} T^{15} + p^{56} T^{16} \)
41 \( 1 - 1324414 T + 1557578476495 T^{2} - 1397128861114137214 T^{3} + \)\(10\!\cdots\!93\)\( T^{4} - \)\(70\!\cdots\!04\)\( T^{5} + \)\(41\!\cdots\!30\)\( T^{6} - \)\(21\!\cdots\!88\)\( T^{7} + \)\(98\!\cdots\!22\)\( T^{8} - \)\(21\!\cdots\!88\)\( p^{7} T^{9} + \)\(41\!\cdots\!30\)\( p^{14} T^{10} - \)\(70\!\cdots\!04\)\( p^{21} T^{11} + \)\(10\!\cdots\!93\)\( p^{28} T^{12} - 1397128861114137214 p^{35} T^{13} + 1557578476495 p^{42} T^{14} - 1324414 p^{49} T^{15} + p^{56} T^{16} \)
43 \( 1 - 2052578 T + 3272237517800 T^{2} - 3543381625624135242 T^{3} + \)\(33\!\cdots\!32\)\( T^{4} - \)\(25\!\cdots\!94\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} - \)\(10\!\cdots\!82\)\( T^{7} + \)\(58\!\cdots\!94\)\( T^{8} - \)\(10\!\cdots\!82\)\( p^{7} T^{9} + \)\(17\!\cdots\!88\)\( p^{14} T^{10} - \)\(25\!\cdots\!94\)\( p^{21} T^{11} + \)\(33\!\cdots\!32\)\( p^{28} T^{12} - 3543381625624135242 p^{35} T^{13} + 3272237517800 p^{42} T^{14} - 2052578 p^{49} T^{15} + p^{56} T^{16} \)
47 \( 1 - 675556 T + 1821019622761 T^{2} - 1275150432927770908 T^{3} + \)\(21\!\cdots\!73\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!94\)\( T^{6} - \)\(94\!\cdots\!28\)\( T^{7} + \)\(98\!\cdots\!82\)\( T^{8} - \)\(94\!\cdots\!28\)\( p^{7} T^{9} + \)\(17\!\cdots\!94\)\( p^{14} T^{10} - \)\(13\!\cdots\!76\)\( p^{21} T^{11} + \)\(21\!\cdots\!73\)\( p^{28} T^{12} - 1275150432927770908 p^{35} T^{13} + 1821019622761 p^{42} T^{14} - 675556 p^{49} T^{15} + p^{56} T^{16} \)
53 \( 1 - 203654 T + 5010478494388 T^{2} - 2726072621558884090 T^{3} + \)\(12\!\cdots\!44\)\( T^{4} - \)\(20\!\cdots\!78\)\( p T^{5} + \)\(20\!\cdots\!32\)\( T^{6} - \)\(21\!\cdots\!86\)\( T^{7} + \)\(26\!\cdots\!90\)\( T^{8} - \)\(21\!\cdots\!86\)\( p^{7} T^{9} + \)\(20\!\cdots\!32\)\( p^{14} T^{10} - \)\(20\!\cdots\!78\)\( p^{22} T^{11} + \)\(12\!\cdots\!44\)\( p^{28} T^{12} - 2726072621558884090 p^{35} T^{13} + 5010478494388 p^{42} T^{14} - 203654 p^{49} T^{15} + p^{56} T^{16} \)
59 \( 1 + 748892 T + 12573581792568 T^{2} + 7506141091634187532 T^{3} + \)\(80\!\cdots\!96\)\( T^{4} + \)\(41\!\cdots\!88\)\( T^{5} + \)\(33\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!64\)\( T^{7} + \)\(99\!\cdots\!46\)\( T^{8} + \)\(15\!\cdots\!64\)\( p^{7} T^{9} + \)\(33\!\cdots\!72\)\( p^{14} T^{10} + \)\(41\!\cdots\!88\)\( p^{21} T^{11} + \)\(80\!\cdots\!96\)\( p^{28} T^{12} + 7506141091634187532 p^{35} T^{13} + 12573581792568 p^{42} T^{14} + 748892 p^{49} T^{15} + p^{56} T^{16} \)
61 \( 1 - 61822 T + 14298093326172 T^{2} + 3793008117290310830 T^{3} + \)\(99\!\cdots\!08\)\( T^{4} + \)\(38\!\cdots\!58\)\( T^{5} + \)\(48\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!50\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} + \)\(16\!\cdots\!50\)\( p^{7} T^{9} + \)\(48\!\cdots\!68\)\( p^{14} T^{10} + \)\(38\!\cdots\!58\)\( p^{21} T^{11} + \)\(99\!\cdots\!08\)\( p^{28} T^{12} + 3793008117290310830 p^{35} T^{13} + 14298093326172 p^{42} T^{14} - 61822 p^{49} T^{15} + p^{56} T^{16} \)
67 \( 1 - 3235604 T + 451536691168 p T^{2} - \)\(10\!\cdots\!84\)\( T^{3} + \)\(49\!\cdots\!84\)\( T^{4} - \)\(16\!\cdots\!32\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!24\)\( T^{7} + \)\(39\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!24\)\( p^{7} T^{9} + \)\(53\!\cdots\!60\)\( p^{14} T^{10} - \)\(16\!\cdots\!32\)\( p^{21} T^{11} + \)\(49\!\cdots\!84\)\( p^{28} T^{12} - \)\(10\!\cdots\!84\)\( p^{35} T^{13} + 451536691168 p^{43} T^{14} - 3235604 p^{49} T^{15} + p^{56} T^{16} \)
71 \( 1 + 4951664 T + 49120129033005 T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!61\)\( T^{4} + \)\(35\!\cdots\!36\)\( T^{5} + \)\(14\!\cdots\!70\)\( T^{6} + \)\(42\!\cdots\!16\)\( T^{7} + \)\(15\!\cdots\!46\)\( T^{8} + \)\(42\!\cdots\!16\)\( p^{7} T^{9} + \)\(14\!\cdots\!70\)\( p^{14} T^{10} + \)\(35\!\cdots\!36\)\( p^{21} T^{11} + \)\(10\!\cdots\!61\)\( p^{28} T^{12} + \)\(19\!\cdots\!44\)\( p^{35} T^{13} + 49120129033005 p^{42} T^{14} + 4951664 p^{49} T^{15} + p^{56} T^{16} \)
73 \( 1 - 11019370 T + 97661917746231 T^{2} - \)\(58\!\cdots\!14\)\( T^{3} + \)\(31\!\cdots\!45\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{5} + \)\(59\!\cdots\!14\)\( T^{6} - \)\(22\!\cdots\!40\)\( T^{7} + \)\(78\!\cdots\!38\)\( T^{8} - \)\(22\!\cdots\!40\)\( p^{7} T^{9} + \)\(59\!\cdots\!14\)\( p^{14} T^{10} - \)\(14\!\cdots\!52\)\( p^{21} T^{11} + \)\(31\!\cdots\!45\)\( p^{28} T^{12} - \)\(58\!\cdots\!14\)\( p^{35} T^{13} + 97661917746231 p^{42} T^{14} - 11019370 p^{49} T^{15} + p^{56} T^{16} \)
79 \( 1 - 4202464 T + 51507191419188 T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!20\)\( T^{4} - \)\(74\!\cdots\!56\)\( T^{5} + \)\(55\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!22\)\( T^{8} - \)\(18\!\cdots\!72\)\( p^{7} T^{9} + \)\(55\!\cdots\!40\)\( p^{14} T^{10} - \)\(74\!\cdots\!56\)\( p^{21} T^{11} + \)\(20\!\cdots\!20\)\( p^{28} T^{12} - \)\(20\!\cdots\!60\)\( p^{35} T^{13} + 51507191419188 p^{42} T^{14} - 4202464 p^{49} T^{15} + p^{56} T^{16} \)
83 \( 1 - 518568 T + 80422420672512 T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(31\!\cdots\!32\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!28\)\( T^{6} - \)\(46\!\cdots\!88\)\( T^{7} + \)\(40\!\cdots\!82\)\( T^{8} - \)\(46\!\cdots\!88\)\( p^{7} T^{9} + \)\(12\!\cdots\!28\)\( p^{14} T^{10} - \)\(15\!\cdots\!56\)\( p^{21} T^{11} + \)\(31\!\cdots\!32\)\( p^{28} T^{12} - \)\(24\!\cdots\!04\)\( p^{35} T^{13} + 80422420672512 p^{42} T^{14} - 518568 p^{49} T^{15} + p^{56} T^{16} \)
89 \( 1 - 4203864 T + 215416282507948 T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!48\)\( T^{4} - \)\(29\!\cdots\!52\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(66\!\cdots\!16\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!16\)\( p^{7} T^{9} + \)\(11\!\cdots\!32\)\( p^{14} T^{10} - \)\(29\!\cdots\!52\)\( p^{21} T^{11} + \)\(19\!\cdots\!48\)\( p^{28} T^{12} - \)\(58\!\cdots\!80\)\( p^{35} T^{13} + 215416282507948 p^{42} T^{14} - 4203864 p^{49} T^{15} + p^{56} T^{16} \)
97 \( 1 - 18621134 T + 638464765638808 T^{2} - \)\(93\!\cdots\!14\)\( T^{3} + \)\(17\!\cdots\!52\)\( T^{4} - \)\(20\!\cdots\!70\)\( T^{5} + \)\(28\!\cdots\!56\)\( T^{6} - \)\(27\!\cdots\!26\)\( T^{7} + \)\(28\!\cdots\!02\)\( T^{8} - \)\(27\!\cdots\!26\)\( p^{7} T^{9} + \)\(28\!\cdots\!56\)\( p^{14} T^{10} - \)\(20\!\cdots\!70\)\( p^{21} T^{11} + \)\(17\!\cdots\!52\)\( p^{28} T^{12} - \)\(93\!\cdots\!14\)\( p^{35} T^{13} + 638464765638808 p^{42} T^{14} - 18621134 p^{49} T^{15} + p^{56} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.39884499895639537174001564447, −6.64304787831468955170627867474, −6.63966979851707807852210296726, −6.37843916801878883056281943083, −6.02389446783938231340212831717, −5.74578234666367604545040406338, −5.72084199112299127057460054060, −5.69094907423072828210647524657, −5.58388356391192602687096004770, −4.95411654581519345186945105889, −4.53632638077800188472590850815, −4.52431823521288325482668208631, −3.98105806524240496348137173170, −3.94216953044909167995387877567, −3.87060184714735040394688769500, −3.65822844024163014916005843345, −3.11629785224852303842550341662, −2.62159998945481565999424197818, −2.27399684565332241808644127608, −1.99628855199381729590217694891, −1.59667404415617968723980830124, −1.43375652739866954972433886572, −1.17395697394488231491743360056, −0.74416317413489961225616394801, −0.46657304302388910805689511729, 0.46657304302388910805689511729, 0.74416317413489961225616394801, 1.17395697394488231491743360056, 1.43375652739866954972433886572, 1.59667404415617968723980830124, 1.99628855199381729590217694891, 2.27399684565332241808644127608, 2.62159998945481565999424197818, 3.11629785224852303842550341662, 3.65822844024163014916005843345, 3.87060184714735040394688769500, 3.94216953044909167995387877567, 3.98105806524240496348137173170, 4.52431823521288325482668208631, 4.53632638077800188472590850815, 4.95411654581519345186945105889, 5.58388356391192602687096004770, 5.69094907423072828210647524657, 5.72084199112299127057460054060, 5.74578234666367604545040406338, 6.02389446783938231340212831717, 6.37843916801878883056281943083, 6.63966979851707807852210296726, 6.64304787831468955170627867474, 7.39884499895639537174001564447

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

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