Properties

Degree $4$
Conductor $2500$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s + 450·9-s + 384·11-s + 256·16-s + 5.48e3·19-s − 1.18e4·29-s − 1.37e4·31-s − 7.20e3·36-s − 756·41-s − 6.14e3·44-s + 1.96e4·49-s + 6.99e4·59-s − 1.96e4·61-s − 4.09e3·64-s + 1.40e5·71-s − 8.76e4·76-s − 9.04e3·79-s + 1.43e5·81-s − 7.69e4·89-s + 1.72e5·99-s + 1.55e5·101-s − 4.13e5·109-s + 1.89e5·116-s − 2.11e5·121-s + 2.19e5·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.85·9-s + 0.956·11-s + 1/4·16-s + 3.48·19-s − 2.60·29-s − 2.56·31-s − 0.925·36-s − 0.0702·41-s − 0.478·44-s + 1.17·49-s + 2.61·59-s − 0.677·61-s − 1/8·64-s + 3.30·71-s − 1.74·76-s − 0.162·79-s + 2.42·81-s − 1.03·89-s + 1.77·99-s + 1.51·101-s − 3.33·109-s + 1.30·116-s − 1.31·121-s + 1.28·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{50} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.55589\)
\(L(\frac12)\) \(\approx\) \(2.55589\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 50 p^{2} T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 19690 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 192 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 480650 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2259070 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 2740 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10420330 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 5910 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6868 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 108239590 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 378 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288092530 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 286503130 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 752228710 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 34980 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 9838 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1563076930 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 70212 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3662758990 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 4520 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4019056190 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 38490 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 17171001790 T^{2} + p^{10} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.52155241674561830901334391163, −14.51085397284922101574563877189, −13.41927511585250737252868425132, −13.33485274730131380284671185393, −12.56495785981300643267951314590, −12.01541310121538520557986715212, −11.36419583327179316600783836124, −10.75092115404869162332093097767, −9.625684455243939828521651815085, −9.613370307717633466875612957780, −9.151728792740791927997915396845, −7.88267787439678049569286541262, −7.14875273808200615221893091120, −7.12274080070210985605437915698, −5.54341723371841183500262925621, −5.22664728751887930531651339036, −3.78151259459389169613567621256, −3.73533592306789052498690499197, −1.76136875721461425397016243766, −0.931167732643556534110499541698, 0.931167732643556534110499541698, 1.76136875721461425397016243766, 3.73533592306789052498690499197, 3.78151259459389169613567621256, 5.22664728751887930531651339036, 5.54341723371841183500262925621, 7.12274080070210985605437915698, 7.14875273808200615221893091120, 7.88267787439678049569286541262, 9.151728792740791927997915396845, 9.613370307717633466875612957780, 9.625684455243939828521651815085, 10.75092115404869162332093097767, 11.36419583327179316600783836124, 12.01541310121538520557986715212, 12.56495785981300643267951314590, 13.33485274730131380284671185393, 13.41927511585250737252868425132, 14.51085397284922101574563877189, 14.52155241674561830901334391163

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