Properties

Degree $4$
Conductor $2980$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s − 3·5-s − 4·7-s + 8·9-s − 4·12-s − 3·13-s + 12·15-s + 16-s + 17-s − 3·19-s − 3·20-s + 16·21-s + 3·23-s + 8·25-s − 12·27-s − 4·28-s − 8·31-s + 12·35-s + 8·36-s − 4·37-s + 12·39-s + 6·41-s − 2·43-s − 24·45-s + 2·47-s − 4·48-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s − 1.34·5-s − 1.51·7-s + 8/3·9-s − 1.15·12-s − 0.832·13-s + 3.09·15-s + 1/4·16-s + 0.242·17-s − 0.688·19-s − 0.670·20-s + 3.49·21-s + 0.625·23-s + 8/5·25-s − 2.30·27-s − 0.755·28-s − 1.43·31-s + 2.02·35-s + 4/3·36-s − 0.657·37-s + 1.92·39-s + 0.937·41-s − 0.304·43-s − 3.57·45-s + 0.291·47-s − 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2980 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2980 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2980\)    =    \(2^{2} \cdot 5 \cdot 149\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2980} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2980,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
149$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T - 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 76 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 3 T - 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−18.5860514542, −17.8382000800, −17.2241633149, −16.8141055416, −16.5263382541, −15.9697928441, −15.7709811752, −14.9773922236, −14.5612088483, −13.2254388791, −12.7775887877, −12.2444167362, −12.0522016129, −11.3114860403, −10.9232383159, −10.4710565113, −9.70868959236, −8.93046945614, −7.72682472869, −7.08779293442, −6.63836760731, −5.93451554981, −5.23669345204, −4.31235997202, −3.22526562957, 0, 3.22526562957, 4.31235997202, 5.23669345204, 5.93451554981, 6.63836760731, 7.08779293442, 7.72682472869, 8.93046945614, 9.70868959236, 10.4710565113, 10.9232383159, 11.3114860403, 12.0522016129, 12.2444167362, 12.7775887877, 13.2254388791, 14.5612088483, 14.9773922236, 15.7709811752, 15.9697928441, 16.5263382541, 16.8141055416, 17.2241633149, 17.8382000800, 18.5860514542

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