Properties

Label 14-21e7-1.1-c29e7-0-0
Degree $14$
Conductor $1801088541$
Sign $1$
Analytic cond. $2.19468\times 10^{14}$
Root an. cond. $10.5775$
Motivic weight $29$
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  + 6.08e4·2-s − 3.34e7·3-s + 1.07e9·4-s − 2.86e9·5-s − 2.03e12·6-s + 4.74e12·7-s − 1.73e13·8-s + 6.40e14·9-s − 1.74e14·10-s + 1.35e14·11-s − 3.59e16·12-s + 1.21e16·13-s + 2.88e17·14-s + 9.58e16·15-s − 9.98e17·16-s − 3.85e17·17-s + 3.89e19·18-s + 7.26e17·19-s − 3.07e18·20-s − 1.58e20·21-s + 8.27e18·22-s + 8.16e19·23-s + 5.81e20·24-s − 8.20e20·25-s + 7.38e20·26-s − 9.19e21·27-s + 5.09e21·28-s + ⋯
L(s)  = 1  + 2.62·2-s − 4.04·3-s + 2.00·4-s − 0.209·5-s − 10.6·6-s + 2.64·7-s − 1.39·8-s + 28/3·9-s − 0.550·10-s + 0.107·11-s − 8.08·12-s + 0.855·13-s + 6.95·14-s + 0.847·15-s − 3.46·16-s − 0.555·17-s + 24.5·18-s + 0.208·19-s − 0.419·20-s − 10.6·21-s + 0.283·22-s + 1.46·23-s + 5.64·24-s − 4.40·25-s + 2.24·26-s − 16.1·27-s + 5.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{7}\right)^{s/2} \, \Gamma_{\C}(s+29/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(3^{7} \cdot 7^{7}\)
Sign: $1$
Analytic conductor: \(2.19468\times 10^{14}\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 3^{7} \cdot 7^{7} ,\ ( \ : [29/2]^{7} ),\ 1 )\)

Particular Values

\(L(15)\) \(\approx\) \(20.56715864\)
\(L(\frac12)\) \(\approx\) \(20.56715864\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{14} T )^{7} \)
7 \( ( 1 - p^{14} T )^{7} \)
good2 \( 1 - 30435 p T + 328882363 p^{3} T^{2} - 604712936295 p^{7} T^{3} + 111487553463921 p^{14} T^{4} - 31117694500814499 p^{20} T^{5} + 14101663836710116145 p^{25} T^{6} - \)\(43\!\cdots\!59\)\( p^{34} T^{7} + 14101663836710116145 p^{54} T^{8} - 31117694500814499 p^{78} T^{9} + 111487553463921 p^{101} T^{10} - 604712936295 p^{123} T^{11} + 328882363 p^{148} T^{12} - 30435 p^{175} T^{13} + p^{203} T^{14} \)
5 \( 1 + 2861618502 T + \)\(16\!\cdots\!03\)\( p T^{2} + \)\(26\!\cdots\!48\)\( p^{2} T^{3} + \)\(10\!\cdots\!97\)\( p^{5} T^{4} - \)\(42\!\cdots\!78\)\( p^{8} T^{5} + \)\(34\!\cdots\!23\)\( p^{12} T^{6} - \)\(48\!\cdots\!12\)\( p^{16} T^{7} + \)\(34\!\cdots\!23\)\( p^{41} T^{8} - \)\(42\!\cdots\!78\)\( p^{66} T^{9} + \)\(10\!\cdots\!97\)\( p^{92} T^{10} + \)\(26\!\cdots\!48\)\( p^{118} T^{11} + \)\(16\!\cdots\!03\)\( p^{146} T^{12} + 2861618502 p^{174} T^{13} + p^{203} T^{14} \)
11 \( 1 - 135879674344284 T + \)\(33\!\cdots\!81\)\( T^{2} - \)\(38\!\cdots\!44\)\( p T^{3} + \)\(44\!\cdots\!61\)\( p^{2} T^{4} - \)\(87\!\cdots\!24\)\( p^{3} T^{5} + \)\(86\!\cdots\!31\)\( p^{5} T^{6} - \)\(84\!\cdots\!84\)\( p^{7} T^{7} + \)\(86\!\cdots\!31\)\( p^{34} T^{8} - \)\(87\!\cdots\!24\)\( p^{61} T^{9} + \)\(44\!\cdots\!61\)\( p^{89} T^{10} - \)\(38\!\cdots\!44\)\( p^{117} T^{11} + \)\(33\!\cdots\!81\)\( p^{145} T^{12} - 135879674344284 p^{174} T^{13} + p^{203} T^{14} \)
13 \( 1 - 933768842195090 p T + \)\(34\!\cdots\!59\)\( p^{2} T^{2} - \)\(47\!\cdots\!20\)\( p^{2} T^{3} + \)\(10\!\cdots\!77\)\( p^{3} T^{4} - \)\(71\!\cdots\!46\)\( p^{5} T^{5} + \)\(91\!\cdots\!55\)\( p^{7} T^{6} - \)\(60\!\cdots\!76\)\( p^{9} T^{7} + \)\(91\!\cdots\!55\)\( p^{36} T^{8} - \)\(71\!\cdots\!46\)\( p^{63} T^{9} + \)\(10\!\cdots\!77\)\( p^{90} T^{10} - \)\(47\!\cdots\!20\)\( p^{118} T^{11} + \)\(34\!\cdots\!59\)\( p^{147} T^{12} - 933768842195090 p^{175} T^{13} + p^{203} T^{14} \)
17 \( 1 + 385869676567281066 T + \)\(26\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(32\!\cdots\!29\)\( T^{4} + \)\(69\!\cdots\!58\)\( p T^{5} + \)\(82\!\cdots\!79\)\( p^{2} T^{6} + \)\(15\!\cdots\!96\)\( p^{3} T^{7} + \)\(82\!\cdots\!79\)\( p^{31} T^{8} + \)\(69\!\cdots\!58\)\( p^{59} T^{9} + \)\(32\!\cdots\!29\)\( p^{87} T^{10} + \)\(10\!\cdots\!52\)\( p^{116} T^{11} + \)\(26\!\cdots\!63\)\( p^{145} T^{12} + 385869676567281066 p^{174} T^{13} + p^{203} T^{14} \)
19 \( 1 - 38214507429662876 p T + \)\(38\!\cdots\!97\)\( T^{2} - \)\(53\!\cdots\!36\)\( T^{3} + \)\(82\!\cdots\!21\)\( T^{4} - \)\(64\!\cdots\!96\)\( p T^{5} + \)\(38\!\cdots\!61\)\( p^{2} T^{6} - \)\(24\!\cdots\!12\)\( p^{3} T^{7} + \)\(38\!\cdots\!61\)\( p^{31} T^{8} - \)\(64\!\cdots\!96\)\( p^{59} T^{9} + \)\(82\!\cdots\!21\)\( p^{87} T^{10} - \)\(53\!\cdots\!36\)\( p^{116} T^{11} + \)\(38\!\cdots\!97\)\( p^{145} T^{12} - 38214507429662876 p^{175} T^{13} + p^{203} T^{14} \)
23 \( 1 - 81623233895572110192 T + \)\(83\!\cdots\!17\)\( T^{2} - \)\(27\!\cdots\!32\)\( p T^{3} + \)\(10\!\cdots\!41\)\( p^{2} T^{4} - \)\(30\!\cdots\!20\)\( p^{3} T^{5} + \)\(84\!\cdots\!85\)\( p^{4} T^{6} - \)\(19\!\cdots\!40\)\( p^{5} T^{7} + \)\(84\!\cdots\!85\)\( p^{33} T^{8} - \)\(30\!\cdots\!20\)\( p^{61} T^{9} + \)\(10\!\cdots\!41\)\( p^{89} T^{10} - \)\(27\!\cdots\!32\)\( p^{117} T^{11} + \)\(83\!\cdots\!17\)\( p^{145} T^{12} - 81623233895572110192 p^{174} T^{13} + p^{203} T^{14} \)
29 \( 1 - \)\(25\!\cdots\!06\)\( T + \)\(15\!\cdots\!07\)\( T^{2} - \)\(25\!\cdots\!04\)\( T^{3} + \)\(92\!\cdots\!21\)\( T^{4} - \)\(11\!\cdots\!38\)\( T^{5} + \)\(32\!\cdots\!83\)\( T^{6} - \)\(31\!\cdots\!56\)\( T^{7} + \)\(32\!\cdots\!83\)\( p^{29} T^{8} - \)\(11\!\cdots\!38\)\( p^{58} T^{9} + \)\(92\!\cdots\!21\)\( p^{87} T^{10} - \)\(25\!\cdots\!04\)\( p^{116} T^{11} + \)\(15\!\cdots\!07\)\( p^{145} T^{12} - \)\(25\!\cdots\!06\)\( p^{174} T^{13} + p^{203} T^{14} \)
31 \( 1 - \)\(30\!\cdots\!52\)\( T + \)\(41\!\cdots\!93\)\( T^{2} - \)\(42\!\cdots\!12\)\( T^{3} + \)\(67\!\cdots\!21\)\( T^{4} + \)\(26\!\cdots\!80\)\( T^{5} + \)\(61\!\cdots\!85\)\( T^{6} + \)\(79\!\cdots\!00\)\( T^{7} + \)\(61\!\cdots\!85\)\( p^{29} T^{8} + \)\(26\!\cdots\!80\)\( p^{58} T^{9} + \)\(67\!\cdots\!21\)\( p^{87} T^{10} - \)\(42\!\cdots\!12\)\( p^{116} T^{11} + \)\(41\!\cdots\!93\)\( p^{145} T^{12} - \)\(30\!\cdots\!52\)\( p^{174} T^{13} + p^{203} T^{14} \)
37 \( 1 - \)\(12\!\cdots\!66\)\( T + \)\(31\!\cdots\!79\)\( p T^{2} + \)\(83\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!37\)\( p T^{4} + \)\(71\!\cdots\!94\)\( T^{5} + \)\(24\!\cdots\!71\)\( T^{6} + \)\(35\!\cdots\!72\)\( T^{7} + \)\(24\!\cdots\!71\)\( p^{29} T^{8} + \)\(71\!\cdots\!94\)\( p^{58} T^{9} + \)\(17\!\cdots\!37\)\( p^{88} T^{10} + \)\(83\!\cdots\!88\)\( p^{116} T^{11} + \)\(31\!\cdots\!79\)\( p^{146} T^{12} - \)\(12\!\cdots\!66\)\( p^{174} T^{13} + p^{203} T^{14} \)
41 \( 1 + \)\(26\!\cdots\!54\)\( T + \)\(95\!\cdots\!91\)\( p T^{2} + \)\(81\!\cdots\!24\)\( T^{3} + \)\(65\!\cdots\!21\)\( T^{4} + \)\(11\!\cdots\!98\)\( T^{5} + \)\(62\!\cdots\!87\)\( T^{6} + \)\(84\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!87\)\( p^{29} T^{8} + \)\(11\!\cdots\!98\)\( p^{58} T^{9} + \)\(65\!\cdots\!21\)\( p^{87} T^{10} + \)\(81\!\cdots\!24\)\( p^{116} T^{11} + \)\(95\!\cdots\!91\)\( p^{146} T^{12} + \)\(26\!\cdots\!54\)\( p^{174} T^{13} + p^{203} T^{14} \)
43 \( 1 + \)\(19\!\cdots\!80\)\( T + \)\(54\!\cdots\!41\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!69\)\( T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(16\!\cdots\!45\)\( T^{6} - \)\(17\!\cdots\!96\)\( T^{7} + \)\(16\!\cdots\!45\)\( p^{29} T^{8} - \)\(17\!\cdots\!36\)\( p^{58} T^{9} + \)\(98\!\cdots\!69\)\( p^{87} T^{10} - \)\(15\!\cdots\!80\)\( p^{116} T^{11} + \)\(54\!\cdots\!41\)\( p^{145} T^{12} + \)\(19\!\cdots\!80\)\( p^{174} T^{13} + p^{203} T^{14} \)
47 \( 1 + \)\(11\!\cdots\!48\)\( T + \)\(12\!\cdots\!65\)\( T^{2} + \)\(81\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!67\)\( p T^{4} + \)\(84\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!65\)\( T^{6} - \)\(34\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!65\)\( p^{29} T^{8} + \)\(84\!\cdots\!40\)\( p^{58} T^{9} + \)\(12\!\cdots\!67\)\( p^{88} T^{10} + \)\(81\!\cdots\!16\)\( p^{116} T^{11} + \)\(12\!\cdots\!65\)\( p^{145} T^{12} + \)\(11\!\cdots\!48\)\( p^{174} T^{13} + p^{203} T^{14} \)
53 \( 1 + \)\(66\!\cdots\!14\)\( T + \)\(45\!\cdots\!35\)\( p T^{2} + \)\(22\!\cdots\!52\)\( T^{3} + \)\(53\!\cdots\!49\)\( T^{4} + \)\(33\!\cdots\!14\)\( T^{5} + \)\(69\!\cdots\!91\)\( T^{6} + \)\(46\!\cdots\!20\)\( T^{7} + \)\(69\!\cdots\!91\)\( p^{29} T^{8} + \)\(33\!\cdots\!14\)\( p^{58} T^{9} + \)\(53\!\cdots\!49\)\( p^{87} T^{10} + \)\(22\!\cdots\!52\)\( p^{116} T^{11} + \)\(45\!\cdots\!35\)\( p^{146} T^{12} + \)\(66\!\cdots\!14\)\( p^{174} T^{13} + p^{203} T^{14} \)
59 \( 1 - \)\(42\!\cdots\!20\)\( p T + \)\(36\!\cdots\!33\)\( T^{2} - \)\(38\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!21\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!05\)\( T^{6} - \)\(64\!\cdots\!56\)\( T^{7} + \)\(12\!\cdots\!05\)\( p^{29} T^{8} - \)\(21\!\cdots\!92\)\( p^{58} T^{9} + \)\(31\!\cdots\!21\)\( p^{87} T^{10} - \)\(38\!\cdots\!40\)\( p^{116} T^{11} + \)\(36\!\cdots\!33\)\( p^{145} T^{12} - \)\(42\!\cdots\!20\)\( p^{175} T^{13} + p^{203} T^{14} \)
61 \( 1 - \)\(28\!\cdots\!86\)\( T + \)\(58\!\cdots\!91\)\( T^{2} - \)\(86\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} - \)\(11\!\cdots\!98\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(85\!\cdots\!68\)\( T^{7} + \)\(10\!\cdots\!03\)\( p^{29} T^{8} - \)\(11\!\cdots\!98\)\( p^{58} T^{9} + \)\(10\!\cdots\!81\)\( p^{87} T^{10} - \)\(86\!\cdots\!76\)\( p^{116} T^{11} + \)\(58\!\cdots\!91\)\( p^{145} T^{12} - \)\(28\!\cdots\!86\)\( p^{174} T^{13} + p^{203} T^{14} \)
67 \( 1 + \)\(21\!\cdots\!12\)\( T + \)\(30\!\cdots\!85\)\( T^{2} - \)\(96\!\cdots\!56\)\( T^{3} + \)\(49\!\cdots\!09\)\( T^{4} - \)\(29\!\cdots\!04\)\( T^{5} + \)\(58\!\cdots\!17\)\( T^{6} - \)\(36\!\cdots\!80\)\( T^{7} + \)\(58\!\cdots\!17\)\( p^{29} T^{8} - \)\(29\!\cdots\!04\)\( p^{58} T^{9} + \)\(49\!\cdots\!09\)\( p^{87} T^{10} - \)\(96\!\cdots\!56\)\( p^{116} T^{11} + \)\(30\!\cdots\!85\)\( p^{145} T^{12} + \)\(21\!\cdots\!12\)\( p^{174} T^{13} + p^{203} T^{14} \)
71 \( 1 - \)\(23\!\cdots\!20\)\( T + \)\(17\!\cdots\!37\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!81\)\( T^{4} - \)\(46\!\cdots\!00\)\( T^{5} + \)\(98\!\cdots\!85\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(98\!\cdots\!85\)\( p^{29} T^{8} - \)\(46\!\cdots\!00\)\( p^{58} T^{9} + \)\(15\!\cdots\!81\)\( p^{87} T^{10} - \)\(16\!\cdots\!20\)\( p^{116} T^{11} + \)\(17\!\cdots\!37\)\( p^{145} T^{12} - \)\(23\!\cdots\!20\)\( p^{174} T^{13} + p^{203} T^{14} \)
73 \( 1 - \)\(14\!\cdots\!06\)\( T + \)\(61\!\cdots\!15\)\( T^{2} - \)\(63\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} - \)\(12\!\cdots\!74\)\( T^{5} + \)\(25\!\cdots\!19\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!19\)\( p^{29} T^{8} - \)\(12\!\cdots\!74\)\( p^{58} T^{9} + \)\(16\!\cdots\!89\)\( p^{87} T^{10} - \)\(63\!\cdots\!48\)\( p^{116} T^{11} + \)\(61\!\cdots\!15\)\( p^{145} T^{12} - \)\(14\!\cdots\!06\)\( p^{174} T^{13} + p^{203} T^{14} \)
79 \( 1 - \)\(44\!\cdots\!48\)\( T + \)\(33\!\cdots\!29\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(58\!\cdots\!61\)\( T^{4} - \)\(83\!\cdots\!64\)\( T^{5} + \)\(78\!\cdots\!97\)\( T^{6} - \)\(10\!\cdots\!56\)\( T^{7} + \)\(78\!\cdots\!97\)\( p^{29} T^{8} - \)\(83\!\cdots\!64\)\( p^{58} T^{9} + \)\(58\!\cdots\!61\)\( p^{87} T^{10} - \)\(36\!\cdots\!12\)\( p^{116} T^{11} + \)\(33\!\cdots\!29\)\( p^{145} T^{12} - \)\(44\!\cdots\!48\)\( p^{174} T^{13} + p^{203} T^{14} \)
83 \( 1 - \)\(47\!\cdots\!76\)\( T + \)\(20\!\cdots\!65\)\( T^{2} - \)\(13\!\cdots\!88\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} - \)\(93\!\cdots\!40\)\( T^{7} + \)\(11\!\cdots\!13\)\( p^{29} T^{8} - \)\(15\!\cdots\!28\)\( p^{58} T^{9} + \)\(19\!\cdots\!09\)\( p^{87} T^{10} - \)\(13\!\cdots\!88\)\( p^{116} T^{11} + \)\(20\!\cdots\!65\)\( p^{145} T^{12} - \)\(47\!\cdots\!76\)\( p^{174} T^{13} + p^{203} T^{14} \)
89 \( 1 - \)\(52\!\cdots\!86\)\( T + \)\(18\!\cdots\!87\)\( T^{2} - \)\(41\!\cdots\!44\)\( T^{3} + \)\(91\!\cdots\!21\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(45\!\cdots\!11\)\( T^{6} - \)\(87\!\cdots\!32\)\( T^{7} + \)\(45\!\cdots\!11\)\( p^{29} T^{8} - \)\(18\!\cdots\!06\)\( p^{58} T^{9} + \)\(91\!\cdots\!21\)\( p^{87} T^{10} - \)\(41\!\cdots\!44\)\( p^{116} T^{11} + \)\(18\!\cdots\!87\)\( p^{145} T^{12} - \)\(52\!\cdots\!86\)\( p^{174} T^{13} + p^{203} T^{14} \)
97 \( 1 - \)\(74\!\cdots\!42\)\( T + \)\(15\!\cdots\!95\)\( T^{2} - \)\(76\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} - \)\(48\!\cdots\!38\)\( T^{5} + \)\(61\!\cdots\!31\)\( T^{6} - \)\(25\!\cdots\!40\)\( T^{7} + \)\(61\!\cdots\!31\)\( p^{29} T^{8} - \)\(48\!\cdots\!38\)\( p^{58} T^{9} + \)\(10\!\cdots\!29\)\( p^{87} T^{10} - \)\(76\!\cdots\!84\)\( p^{116} T^{11} + \)\(15\!\cdots\!95\)\( p^{145} T^{12} - \)\(74\!\cdots\!42\)\( p^{174} T^{13} + p^{203} T^{14} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.69055018126854644800165922638, −4.57661723860691897384767003076, −4.54634336422261425702059346578, −4.38840481827351231281155974356, −3.93674573252846255626781716272, −3.75474658681915674625014110675, −3.62925752638482997596555375395, −3.60027104604450395996097407959, −3.55603961952400350205658365785, −3.49959256719699553358464537279, −2.57797784343392557444159049568, −2.56007218505302150080745515644, −2.29514670413861352688568348528, −2.13100959655409350593330137099, −2.05065419087509436915790924905, −1.72202696557222229566760157688, −1.63567414241960552305670038608, −1.28960731849444808813443017682, −1.19724159682022932055312109468, −0.866817809499206927815907983602, −0.813195138188114255083487529122, −0.58047337317106547825876235140, −0.49929271559305763619656189077, −0.49803418075559936191300793391, −0.22206477937588874911561916746, 0.22206477937588874911561916746, 0.49803418075559936191300793391, 0.49929271559305763619656189077, 0.58047337317106547825876235140, 0.813195138188114255083487529122, 0.866817809499206927815907983602, 1.19724159682022932055312109468, 1.28960731849444808813443017682, 1.63567414241960552305670038608, 1.72202696557222229566760157688, 2.05065419087509436915790924905, 2.13100959655409350593330137099, 2.29514670413861352688568348528, 2.56007218505302150080745515644, 2.57797784343392557444159049568, 3.49959256719699553358464537279, 3.55603961952400350205658365785, 3.60027104604450395996097407959, 3.62925752638482997596555375395, 3.75474658681915674625014110675, 3.93674573252846255626781716272, 4.38840481827351231281155974356, 4.54634336422261425702059346578, 4.57661723860691897384767003076, 4.69055018126854644800165922638

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

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