Properties

Degree 4
Conductor $ 3 \cdot 5 \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 − 4-s − 3·5-s + 7-s + 8-s − 2·9-s + 3·10-s + 3·11-s + 4·13-s − 14-s − 16-s + 2·18-s − 6·19-s + 3·20-s − 3·22-s + 8·25-s − 4·26-s + 3·27-s − 28-s − 2·29-s − 6·31-s + 5·32-s − 3·35-s + 2·36-s − 3·37-s + 6·38-s − 3·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.904·11-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 0.471·18-s − 1.37·19-s + 0.670·20-s − 0.639·22-s + 8/5·25-s − 0.784·26-s + 0.577·27-s − 0.188·28-s − 0.371·29-s − 1.07·31-s + 0.883·32-s − 0.507·35-s + 1/3·36-s − 0.493·37-s + 0.973·38-s − 0.474·40-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(555\)    =    \(3 \cdot 5 \cdot 37\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{555} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 555,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2569247202$
$L(\frac12)$  $\approx$  $0.2569247202$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;37\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;37\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$V_4$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$V_4$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 13 T + 148 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−19.7228696001, −19.2625330172, −18.8141373867, −18.1703438031, −17.7245770549, −16.8683974252, −16.6349961675, −15.7100777738, −15.175760324, −14.4847639708, −13.9722591121, −13.0048256217, −12.3459379241, −11.4809073364, −11.1342787764, −10.3300672221, −8.98676232797, −8.78127121246, −8.19059310857, −7.14849199777, −6.11392088356, −4.60763755176, −3.67549431481, 3.67549431481, 4.60763755176, 6.11392088356, 7.14849199777, 8.19059310857, 8.78127121246, 8.98676232797, 10.3300672221, 11.1342787764, 11.4809073364, 12.3459379241, 13.0048256217, 13.9722591121, 14.4847639708, 15.175760324, 15.7100777738, 16.6349961675, 16.8683974252, 17.7245770549, 18.1703438031, 18.8141373867, 19.2625330172, 19.7228696001

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