Properties

Degree 4
Conductor $ 5^{2} \cdot 37 $
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 + 6-s + 7-s − 3·8-s − 9-s − 11-s − 12-s − 2·13-s − 14-s + 16-s + 2·17-s + 18-s − 21-s + 22-s − 4·23-s + 3·24-s + 25-s + 2·26-s + 28-s + 2·29-s + 10·31-s + 32-s + 33-s − 2·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 1.06·8-s − 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s + 1.79·31-s + 0.176·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(925\)    =    \(5^{2} \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{925} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 925,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3262333436$
$L(\frac12)$  $\approx$  $0.3262333436$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;37\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
41$D_{4}$ \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 100 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 11 T + 118 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−19.9585653344, −19.687522678, −18.8022141826, −18.3971426664, −17.8838377117, −17.3252037336, −16.9767987929, −16.2232503986, −15.6938447794, −15.0691342122, −14.469470917, −13.7811975793, −12.9358793822, −12.063341508, −11.7393336972, −11.1967724964, −10.180356712, −9.86128598838, −8.83144962618, −8.22237526025, −7.40817978324, −6.37913670748, −5.72530814979, −4.57934574593, −2.76893362756, 2.76893362756, 4.57934574593, 5.72530814979, 6.37913670748, 7.40817978324, 8.22237526025, 8.83144962618, 9.86128598838, 10.180356712, 11.1967724964, 11.7393336972, 12.063341508, 12.9358793822, 13.7811975793, 14.469470917, 15.0691342122, 15.6938447794, 16.2232503986, 16.9767987929, 17.3252037336, 17.8838377117, 18.3971426664, 18.8022141826, 19.687522678, 19.9585653344

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