Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{5} $
Sign $1$
Motivic weight 1
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

\( d \)  =  \(4\)
\( N \)  =  \(12500\)    =    \(2^{2} \cdot 5^{5}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{12500} (1, \cdot )$
Sato-Tate  :  $N(G_{3,3})$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 12500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.000455522$
$L(\frac12)$  $\approx$  $1.000455522$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good3$V_4$ \( 1 + p^{2} T^{4} \)
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$V_4$ \( 1 + p^{2} T^{4} \)
17$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$D_4$ \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$V_4$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
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$V_4$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
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$V_4$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
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$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$V_4$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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, −10.160141098, −9.5492928378, −9.1231458947, −8.6602377903, −7.8358810525, −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, 7.8358810525, 8.6602377903, 9.1231458947, 9.5492928378, 10.160141098, 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