Properties

Degree 4
Conductor $ 241 \cdot 269 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s − 5-s + 2·6-s − 7-s + 3·8-s + 2·9-s − 10-s + 6·11-s + 2·12-s − 13-s − 14-s − 2·15-s + 16-s − 2·17-s + 2·18-s + 3·19-s − 20-s − 2·21-s + 6·22-s − 2·23-s + 6·24-s − 6·25-s − 26-s + 6·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1.06·8-s + 2/3·9-s − 0.316·10-s + 1.80·11-s + 0.577·12-s − 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.471·18-s + 0.688·19-s − 0.223·20-s − 0.436·21-s + 1.27·22-s − 0.417·23-s + 1.22·24-s − 6/5·25-s − 0.196·26-s + 1.15·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64829 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64829 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64829\)    =    \(241 \cdot 269\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{64829} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 64829,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.186146820$
$L(\frac12)$  $\approx$  $3.186146820$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{241,\;269\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{241,\;269\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad241$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 13 T + p T^{2} ) \)
269$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 8 T + p T^{2} ) \)
good2$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 55 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 13 T + 111 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T - 74 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T - 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 94 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 9 T + 185 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−14.3812549436, −14.0609807472, −13.7129517020, −13.2325865607, −12.9287955211, −12.1355400263, −11.8782096740, −11.5204616076, −10.9466540353, −10.3111028712, −9.83279896067, −9.27154380142, −8.94169572041, −8.44346236296, −7.61836652117, −7.46919470394, −6.82218605209, −6.34935089590, −5.66630300706, −4.77717543929, −4.21591937213, −3.77389303943, −3.26264667312, −2.29812237305, −1.54485497860, 1.54485497860, 2.29812237305, 3.26264667312, 3.77389303943, 4.21591937213, 4.77717543929, 5.66630300706, 6.34935089590, 6.82218605209, 7.46919470394, 7.61836652117, 8.44346236296, 8.94169572041, 9.27154380142, 9.83279896067, 10.3111028712, 10.9466540353, 11.5204616076, 11.8782096740, 12.1355400263, 12.9287955211, 13.2325865607, 13.7129517020, 14.0609807472, 14.3812549436

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