Dirichlet series
L(s) = 1 | − 4.19e6·2-s − 1.29e10·3-s + 1.31e13·4-s − 3.98e14·5-s + 5.44e16·6-s + 1.17e18·7-s − 3.68e19·8-s − 4.86e20·9-s + 1.67e21·10-s − 1.16e22·11-s − 1.71e23·12-s − 1.65e24·13-s − 4.92e24·14-s + 5.17e24·15-s + 9.67e25·16-s + 3.43e26·17-s + 2.04e27·18-s − 5.26e26·19-s − 5.26e27·20-s − 1.52e28·21-s + 4.87e28·22-s + 2.29e29·23-s + 4.78e29·24-s + 1.34e30·25-s + 6.95e30·26-s + 1.05e31·27-s + 1.55e31·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.716·3-s + 3/2·4-s − 0.373·5-s + 1.01·6-s + 0.795·7-s − 1.41·8-s − 1.48·9-s + 0.528·10-s − 0.473·11-s − 1.07·12-s − 1.86·13-s − 1.12·14-s + 0.267·15-s + 5/4·16-s + 1.20·17-s + 2.09·18-s − 0.169·19-s − 0.560·20-s − 0.569·21-s + 0.669·22-s + 1.21·23-s + 1.01·24-s + 1.18·25-s + 2.63·26-s + 1.77·27-s + 1.19·28-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(4\) = \(2^{2}\) |
Sign: | $1$ |
Analytic conductor: | \(548.593\) |
Root analytic conductor: | \(4.83963\) |
Motivic weight: | \(43\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((4,\ 4,\ (\ :43/2, 43/2),\ 1)\) |
Particular Values
\(L(22)\) | \(\approx\) | \(0.8403225686\) |
\(L(\frac12)\) | \(\approx\) | \(0.8403225686\) |
\(L(\frac{45}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_1$ | \( ( 1 + p^{21} T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + 4327210328 p T + 299551664144015314 p^{7} T^{2} + 4327210328 p^{44} T^{3} + p^{86} T^{4} \) |
5 | $D_{4}$ | \( 1 + 637860338052 p^{4} T - \)\(24\!\cdots\!58\)\( p^{11} T^{2} + 637860338052 p^{47} T^{3} + p^{86} T^{4} \) | |
7 | $D_{4}$ | \( 1 - 1174870033543241008 T + \)\(91\!\cdots\!14\)\( p^{3} T^{2} - 1174870033543241008 p^{43} T^{3} + p^{86} T^{4} \) | |
11 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!16\)\( p T + \)\(32\!\cdots\!86\)\( p^{2} T^{2} + \)\(10\!\cdots\!16\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
13 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!68\)\( p T + \)\(74\!\cdots\!78\)\( p^{4} T^{2} + \)\(12\!\cdots\!68\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
17 | $D_{4}$ | \( 1 - \)\(20\!\cdots\!04\)\( p T + \)\(32\!\cdots\!14\)\( p^{3} T^{2} - \)\(20\!\cdots\!04\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
19 | $D_{4}$ | \( 1 + \)\(52\!\cdots\!00\)\( T + \)\(65\!\cdots\!22\)\( p T^{2} + \)\(52\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
23 | $D_{4}$ | \( 1 - \)\(22\!\cdots\!36\)\( T + \)\(31\!\cdots\!46\)\( p T^{2} - \)\(22\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
29 | $D_{4}$ | \( 1 - \)\(11\!\cdots\!80\)\( p T + \)\(19\!\cdots\!58\)\( p^{2} T^{2} - \)\(11\!\cdots\!80\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
31 | $D_{4}$ | \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(10\!\cdots\!06\)\( p T^{2} - \)\(16\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(21\!\cdots\!98\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
41 | $D_{4}$ | \( 1 - \)\(17\!\cdots\!64\)\( p T + \)\(34\!\cdots\!06\)\( p^{2} T^{2} - \)\(17\!\cdots\!64\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(23\!\cdots\!36\)\( T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
47 | $D_{4}$ | \( 1 + \)\(39\!\cdots\!92\)\( T + \)\(27\!\cdots\!62\)\( T^{2} + \)\(39\!\cdots\!92\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
53 | $D_{4}$ | \( 1 - \)\(21\!\cdots\!56\)\( T + \)\(24\!\cdots\!38\)\( T^{2} - \)\(21\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
59 | $D_{4}$ | \( 1 - \)\(67\!\cdots\!40\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(67\!\cdots\!40\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
61 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(94\!\cdots\!66\)\( T^{2} - \)\(10\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
67 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!52\)\( T + \)\(36\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!52\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
71 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!64\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(10\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
73 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
79 | $D_{4}$ | \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(90\!\cdots\!78\)\( T^{2} + \)\(66\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
83 | $D_{4}$ | \( 1 - \)\(54\!\cdots\!56\)\( T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(54\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
89 | $D_{4}$ | \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(85\!\cdots\!38\)\( T^{2} - \)\(29\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
97 | $D_{4}$ | \( 1 + \)\(22\!\cdots\!12\)\( T + \)\(34\!\cdots\!82\)\( T^{2} + \)\(22\!\cdots\!12\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−17.96160645448336421532711381868, −17.33080688422186730237448576095, −16.97653324333278657278071332348, −16.09370194834217472692218438686, −14.80832250795167993497721456437, −14.36971796286312416058226775268, −12.28126007326664453601815168481, −11.90841038740923063955197932070, −10.94928811191908143949691785540, −10.37818656545802728097423779032, −9.193784287915135026032990493737, −8.295072631434559773077479830200, −7.65723300750753107664232097309, −6.68061998450529215251991083419, −5.47618812107195880219590385620, −4.86185530703986246145813127644, −2.86481725201026409060499933236, −2.50392942226333406092493706109, −0.878088391737027664620773659112, −0.58129219454947557179969273093, 0.58129219454947557179969273093, 0.878088391737027664620773659112, 2.50392942226333406092493706109, 2.86481725201026409060499933236, 4.86185530703986246145813127644, 5.47618812107195880219590385620, 6.68061998450529215251991083419, 7.65723300750753107664232097309, 8.295072631434559773077479830200, 9.193784287915135026032990493737, 10.37818656545802728097423779032, 10.94928811191908143949691785540, 11.90841038740923063955197932070, 12.28126007326664453601815168481, 14.36971796286312416058226775268, 14.80832250795167993497721456437, 16.09370194834217472692218438686, 16.97653324333278657278071332348, 17.33080688422186730237448576095, 17.96160645448336421532711381868