Properties

Degree 4
Conductor $ 2^{3} \cdot 59 $
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 − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s − 9-s + 10-s − 2·11-s + 12-s + 4·13-s + 14-s + 15-s − 16-s + 18-s + 3·19-s + 20-s + 21-s + 2·22-s − 4·23-s − 3·24-s + 25-s − 4·26-s + 28-s + 29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.688·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(472\)    =    \(2^{3} \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{472} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 472,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2274474476$
$L(\frac12)$  $\approx$  $0.2274474476$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;59\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + p T^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 12 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
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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

−19.9946934732, −19.6099856688, −18.8456701559, −18.4015053937, −17.9008354561, −17.3909465659, −16.6033291554, −16.118618406, −15.7055726147, −14.7117385888, −13.8833193799, −13.3753015888, −12.6542563166, −11.7832194413, −11.1174776978, −10.4261165663, −9.72280446813, −8.80136135476, −8.16425698614, −7.31419317654, −6.0921293012, −5.11015045889, −3.7289209158, 3.7289209158, 5.11015045889, 6.0921293012, 7.31419317654, 8.16425698614, 8.80136135476, 9.72280446813, 10.4261165663, 11.1174776978, 11.7832194413, 12.6542563166, 13.3753015888, 13.8833193799, 14.7117385888, 15.7055726147, 16.118618406, 16.6033291554, 17.3909465659, 17.9008354561, 18.4015053937, 18.8456701559, 19.6099856688, 19.9946934732

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