Properties

Degree 4
Conductor $ 5 \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  − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s + 7-s + 9-s + 4·10-s + 2·11-s − 12-s − 2·13-s − 2·14-s + 2·15-s + 16-s − 2·18-s + 19-s − 2·20-s − 21-s − 4·22-s + 2·23-s − 2·25-s + 4·26-s − 4·27-s + 28-s − 9·29-s − 4·30-s + 4·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.417·23-s − 2/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.67·29-s − 0.730·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(295\)    =    \(5 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{295} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 295,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1492683688$
$L(\frac12)$  $\approx$  $0.1492683688$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;59\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;59\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
67$V_4$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$V_4$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 110 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$V_4$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−19.5943790917, −19.1705597141, −18.6253981472, −18.1063434879, −17.3428767194, −17.0651471551, −16.4697340363, −15.4409573542, −15.1670527978, −14.1846862585, −13.2469970219, −12.2291821794, −11.6895301751, −11.0785083624, −10.0882154598, −9.41053737416, −8.6522401113, −7.78228122162, −7.06932004114, −5.61010872777, −4.12033228144, 4.12033228144, 5.61010872777, 7.06932004114, 7.78228122162, 8.6522401113, 9.41053737416, 10.0882154598, 11.0785083624, 11.6895301751, 12.2291821794, 13.2469970219, 14.1846862585, 15.1670527978, 15.4409573542, 16.4697340363, 17.0651471551, 17.3428767194, 18.1063434879, 18.6253981472, 19.1705597141, 19.5943790917

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