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Properties

Label 4-10e2-1.1-c31e2-0-1
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $3706.02$
Root an. cond. $7.80237$
Motivic weight $31$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.55e4·2-s − 5.90e6·3-s + 3.22e9·4-s + 6.10e10·5-s + 3.86e11·6-s + 1.80e13·7-s − 1.40e14·8-s − 6.39e14·9-s − 4.00e15·10-s − 1.74e16·11-s − 1.90e16·12-s − 2.54e16·13-s − 1.18e18·14-s − 3.60e17·15-s + 5.76e18·16-s − 1.63e19·17-s + 4.19e19·18-s − 1.07e20·19-s + 1.96e20·20-s − 1.06e20·21-s + 1.14e21·22-s − 5.77e20·23-s + 8.31e20·24-s + 2.79e21·25-s + 1.67e21·26-s + 4.11e21·27-s + 5.80e22·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.237·3-s + 3/2·4-s + 0.894·5-s + 0.335·6-s + 1.43·7-s − 1.41·8-s − 1.03·9-s − 1.26·10-s − 1.25·11-s − 0.356·12-s − 0.138·13-s − 2.02·14-s − 0.212·15-s + 5/4·16-s − 1.38·17-s + 1.46·18-s − 1.63·19-s + 1.34·20-s − 0.340·21-s + 1.77·22-s − 0.451·23-s + 0.335·24-s + 3/5·25-s + 0.195·26-s + 0.268·27-s + 2.15·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(3706.02\)
Root analytic conductor: \(7.80237\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{33}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{15} T )^{2} \)
5$C_1$ \( ( 1 - p^{15} T )^{2} \)
good3$D_{4}$ \( 1 + 656084 p^{2} T + 308501155394 p^{7} T^{2} + 656084 p^{33} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 - 2573795013316 p T + \)\(13\!\cdots\!22\)\( p^{4} T^{2} - 2573795013316 p^{32} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 17416121431336176 T + \)\(35\!\cdots\!46\)\( p^{2} T^{2} + 17416121431336176 p^{31} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 + 1960754787185492 p T + \)\(23\!\cdots\!74\)\( p^{3} T^{2} + 1960754787185492 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 + 16396230747963647148 T + \)\(13\!\cdots\!26\)\( p T^{2} + 16396230747963647148 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 5677916680435004360 p T + \)\(21\!\cdots\!58\)\( p^{2} T^{2} + 5677916680435004360 p^{32} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 + \)\(57\!\cdots\!36\)\( T - \)\(61\!\cdots\!14\)\( p T^{2} + \)\(57\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 - \)\(67\!\cdots\!40\)\( T + \)\(15\!\cdots\!02\)\( p T^{2} - \)\(67\!\cdots\!40\)\( p^{31} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 - \)\(56\!\cdots\!64\)\( T + \)\(21\!\cdots\!86\)\( T^{2} - \)\(56\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 - \)\(28\!\cdots\!32\)\( T + \)\(10\!\cdots\!82\)\( T^{2} - \)\(28\!\cdots\!32\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 - \)\(18\!\cdots\!84\)\( T + \)\(14\!\cdots\!46\)\( T^{2} - \)\(18\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(25\!\cdots\!84\)\( T + \)\(20\!\cdots\!46\)\( p T^{2} - \)\(25\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 + \)\(20\!\cdots\!28\)\( T + \)\(20\!\cdots\!02\)\( T^{2} + \)\(20\!\cdots\!28\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(93\!\cdots\!44\)\( T + \)\(56\!\cdots\!78\)\( T^{2} - \)\(93\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 - \)\(68\!\cdots\!20\)\( p T + \)\(12\!\cdots\!18\)\( T^{2} - \)\(68\!\cdots\!20\)\( p^{32} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 + \)\(52\!\cdots\!76\)\( T + \)\(39\!\cdots\!66\)\( T^{2} + \)\(52\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 - \)\(35\!\cdots\!52\)\( T + \)\(23\!\cdots\!42\)\( T^{2} - \)\(35\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 + \)\(15\!\cdots\!56\)\( T + \)\(10\!\cdots\!26\)\( T^{2} + \)\(15\!\cdots\!56\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(15\!\cdots\!78\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(10\!\cdots\!60\)\( T + \)\(11\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 + \)\(16\!\cdots\!76\)\( T + \)\(62\!\cdots\!78\)\( T^{2} + \)\(16\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 - \)\(59\!\cdots\!20\)\( T + \)\(48\!\cdots\!78\)\( T^{2} - \)\(59\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(83\!\cdots\!72\)\( T + \)\(93\!\cdots\!02\)\( T^{2} - \)\(83\!\cdots\!72\)\( p^{31} T^{3} + p^{62} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.36071533348091776094050780466, −12.77433843909785573955697051680, −11.47537877094027112145931539672, −11.40245634256205334941756820671, −10.45006599346610319078834593893, −10.26924689275884396820989377421, −9.131936585573792538904487484507, −8.481301101531445725476991858886, −8.213945457805781123827835595005, −7.41210979656458481029895846295, −6.25248415101165197798687948959, −6.08891750021286976143166536811, −4.95846014527165799493772555322, −4.49057038408816486893496607961, −2.73285973627266801187767919575, −2.50451788042048994607758862042, −1.81074769645060792390397805515, −1.15784275083442248784408547920, 0, 0, 1.15784275083442248784408547920, 1.81074769645060792390397805515, 2.50451788042048994607758862042, 2.73285973627266801187767919575, 4.49057038408816486893496607961, 4.95846014527165799493772555322, 6.08891750021286976143166536811, 6.25248415101165197798687948959, 7.41210979656458481029895846295, 8.213945457805781123827835595005, 8.481301101531445725476991858886, 9.131936585573792538904487484507, 10.26924689275884396820989377421, 10.45006599346610319078834593893, 11.40245634256205334941756820671, 11.47537877094027112145931539672, 12.77433843909785573955697051680, 13.36071533348091776094050780466

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