Properties

Degree 4
Conductor $ 23 \cdot 257 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 2·7-s − 3·8-s + 3·9-s + 12·10-s − 5·11-s − 12·12-s − 4·13-s + 6·14-s + 12·15-s + 3·16-s − 3·17-s − 9·18-s − 4·19-s − 16·20-s + 6·21-s + 15·22-s − 2·23-s + 9·24-s + 6·25-s + 12·26-s − 8·28-s + 4·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 0.755·7-s − 1.06·8-s + 9-s + 3.79·10-s − 1.50·11-s − 3.46·12-s − 1.10·13-s + 1.60·14-s + 3.09·15-s + 3/4·16-s − 0.727·17-s − 2.12·18-s − 0.917·19-s − 3.57·20-s + 1.30·21-s + 3.19·22-s − 0.417·23-s + 1.83·24-s + 6/5·25-s + 2.35·26-s − 1.51·28-s + 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5911 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5911 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5911\)    =    \(23 \cdot 257\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5911} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 5911,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{23,\;257\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{23,\;257\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad23$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
257$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$D_{4}$ \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 86 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 5 T + 70 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 128 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−17.7701686365, −17.2638803735, −17.0746572888, −16.2817988126, −16.1892176479, −15.6051090264, −15.2664157109, −14.5423169327, −13.4277415279, −12.6814251129, −12.3276684413, −11.8350722648, −11.2484327000, −10.8034282512, −10.4238496413, −9.78759162464, −9.24574736632, −8.34539623021, −7.88326357762, −7.63845454671, −6.63642716590, −6.13898430263, −5.04885371662, −4.38589172029, −2.89756848057, 0, 0, 2.89756848057, 4.38589172029, 5.04885371662, 6.13898430263, 6.63642716590, 7.63845454671, 7.88326357762, 8.34539623021, 9.24574736632, 9.78759162464, 10.4238496413, 10.8034282512, 11.2484327000, 11.8350722648, 12.3276684413, 12.6814251129, 13.4277415279, 14.5423169327, 15.2664157109, 15.6051090264, 16.1892176479, 16.2817988126, 17.0746572888, 17.2638803735, 17.7701686365

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