Properties

Degree $4$
Conductor $7549$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 7·7-s − 3·8-s + 9-s + 9·10-s − 6·11-s − 8·12-s + 2·13-s + 21·14-s + 6·15-s + 3·16-s − 2·17-s − 3·18-s − 19-s − 12·20-s + 14·21-s + 18·22-s − 23-s + 6·24-s + 4·25-s − 6·26-s − 2·27-s − 28·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 2.64·7-s − 1.06·8-s + 1/3·9-s + 2.84·10-s − 1.80·11-s − 2.30·12-s + 0.554·13-s + 5.61·14-s + 1.54·15-s + 3/4·16-s − 0.485·17-s − 0.707·18-s − 0.229·19-s − 2.68·20-s + 3.05·21-s + 3.83·22-s − 0.208·23-s + 1.22·24-s + 4/5·25-s − 1.17·26-s − 0.384·27-s − 5.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7549\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7549} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7549,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad7549$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 148 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 62 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 23 T + 264 T^{2} - 23 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 145 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 13 T + 167 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.6233326206, −16.8133130710, −16.5649884160, −16.1074698367, −15.8783710153, −15.5109459209, −14.9266677000, −13.6384825781, −13.0815140719, −12.7666062662, −12.1823939437, −11.6233289252, −10.9026823072, −10.4738312018, −10.2020774276, −9.44023370596, −9.11497988036, −8.29184524629, −7.86950415435, −7.20654689735, −6.49212146811, −6.02827429272, −5.07765493599, −3.71332903296, −2.97641785817, 0, 0, 2.97641785817, 3.71332903296, 5.07765493599, 6.02827429272, 6.49212146811, 7.20654689735, 7.86950415435, 8.29184524629, 9.11497988036, 9.44023370596, 10.2020774276, 10.4738312018, 10.9026823072, 11.6233289252, 12.1823939437, 12.7666062662, 13.0815140719, 13.6384825781, 14.9266677000, 15.5109459209, 15.8783710153, 16.1074698367, 16.5649884160, 16.8133130710, 17.6233326206

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