Properties

Degree 4
Conductor $ 2 \cdot 23117 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4-s − 6·5-s + 8·6-s − 4·7-s + 2·8-s + 7·9-s + 12·10-s − 7·11-s − 4·12-s − 7·13-s + 8·14-s + 24·15-s − 3·16-s − 17-s − 14·18-s − 4·19-s − 6·20-s + 16·21-s + 14·22-s − 2·23-s − 8·24-s + 19·25-s + 14·26-s − 4·27-s − 4·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.68·5-s + 3.26·6-s − 1.51·7-s + 0.707·8-s + 7/3·9-s + 3.79·10-s − 2.11·11-s − 1.15·12-s − 1.94·13-s + 2.13·14-s + 6.19·15-s − 3/4·16-s − 0.242·17-s − 3.29·18-s − 0.917·19-s − 1.34·20-s + 3.49·21-s + 2.98·22-s − 0.417·23-s − 1.63·24-s + 19/5·25-s + 2.74·26-s − 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 46234 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 46234 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(46234\)    =    \(2 \cdot 23117\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{46234} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 46234,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23117\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;23117\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
23117$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 200 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 53 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$D_{4}$ \( 1 - 8 T + 4 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 7 T + T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T - 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 22 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−15.6793869195, −15.5168826626, −15.0393091864, −14.3134051489, −13.3687166354, −12.6583702437, −12.54175903, −12.2795026088, −11.7695952259, −11.225023383, −10.7737929161, −10.5085643553, −10.188380241, −9.52593292253, −8.8203751527, −8.28873035344, −7.61478704025, −7.40141515679, −7.06163721341, −6.25901498483, −5.4797124067, −4.97858294024, −4.4760198566, −3.67112207875, −2.72297767378, 0, 0, 0, 2.72297767378, 3.67112207875, 4.4760198566, 4.97858294024, 5.4797124067, 6.25901498483, 7.06163721341, 7.40141515679, 7.61478704025, 8.28873035344, 8.8203751527, 9.52593292253, 10.188380241, 10.5085643553, 10.7737929161, 11.225023383, 11.7695952259, 12.2795026088, 12.54175903, 12.6583702437, 13.3687166354, 14.3134051489, 15.0393091864, 15.5168826626, 15.6793869195

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