Properties

Label 4-12500-1.1-c1e2-0-0
Degree $4$
Conductor $12500$
Sign $1$
Analytic cond. $0.797010$
Root an. cond. $0.944856$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·11-s + 16-s + 5·19-s + 5·29-s − 31-s − 41-s − 4·44-s − 10·49-s − 15·59-s − 6·61-s − 64-s − 11·71-s − 5·76-s + 10·79-s − 9·81-s + 20·89-s + 101-s + 103-s + 107-s + 109-s + 113-s − 5·116-s − 10·121-s + 124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.20·11-s + 1/4·16-s + 1.14·19-s + 0.928·29-s − 0.179·31-s − 0.156·41-s − 0.603·44-s − 1.42·49-s − 1.95·59-s − 0.768·61-s − 1/8·64-s − 1.30·71-s − 0.573·76-s + 1.12·79-s − 81-s + 2.11·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.464·116-s − 0.909·121-s + 0.0898·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12500\)    =    \(2^{2} \cdot 5^{5}\)
Sign: $1$
Analytic conductor: \(0.797010\)
Root analytic conductor: \(0.944856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.000455522\)
\(L(\frac12)\) \(\approx\) \(1.000455522\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 15 T + 143 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
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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

−16.3486478295, −15.9168177788, −15.2498750496, −14.8387821617, −14.2448627481, −13.9071977484, −13.5499526755, −12.7792910762, −12.3928787507, −11.7248272699, −11.5232216023, −10.7002898501, −10.1601410980, −9.54929283780, −9.12314589470, −8.66023779030, −7.83588105250, −7.41712954412, −6.49364126284, −6.13939202948, −5.15084205654, −4.60495208272, −3.74290124925, −2.99781820026, −1.42931298829, 1.42931298829, 2.99781820026, 3.74290124925, 4.60495208272, 5.15084205654, 6.13939202948, 6.49364126284, 7.41712954412, 7.83588105250, 8.66023779030, 9.12314589470, 9.54929283780, 10.1601410980, 10.7002898501, 11.5232216023, 11.7248272699, 12.3928787507, 12.7792910762, 13.5499526755, 13.9071977484, 14.2448627481, 14.8387821617, 15.2498750496, 15.9168177788, 16.3486478295

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