Properties

Degree 4
Conductor $ 2^{5} \cdot 5 \cdot 113 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s + 4-s − 3·5-s + 4·6-s − 4·7-s − 8-s + 8·9-s + 3·10-s − 4·11-s − 4·12-s − 6·13-s + 4·14-s + 12·15-s + 16-s − 2·17-s − 8·18-s − 3·20-s + 16·21-s + 4·22-s − 2·23-s + 4·24-s + 2·25-s + 6·26-s − 12·27-s − 4·28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 1.63·6-s − 1.51·7-s − 0.353·8-s + 8/3·9-s + 0.948·10-s − 1.20·11-s − 1.15·12-s − 1.66·13-s + 1.06·14-s + 3.09·15-s + 1/4·16-s − 0.485·17-s − 1.88·18-s − 0.670·20-s + 3.49·21-s + 0.852·22-s − 0.417·23-s + 0.816·24-s + 2/5·25-s + 1.17·26-s − 2.30·27-s − 0.755·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(18080\)    =    \(2^{5} \cdot 5 \cdot 113\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{18080} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 18080,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;113\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;113\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
113$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 14 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 52 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 92 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T - 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_4$ \( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−16.3515682839, −16.0010833644, −15.7684513703, −15.2911501434, −14.7838884289, −13.8393624636, −13.1212673911, −12.5007858232, −12.3293222118, −12.0550574465, −11.4057042384, −10.849039931, −10.6845459998, −10.04674646, −9.54134315312, −8.97217647354, −7.8663174656, −7.5118669482, −7.06192903296, −6.40047824148, −5.83619071717, −5.25681546353, −4.58339964474, −3.67231091169, −2.57410547876, 0, 0, 2.57410547876, 3.67231091169, 4.58339964474, 5.25681546353, 5.83619071717, 6.40047824148, 7.06192903296, 7.5118669482, 7.8663174656, 8.97217647354, 9.54134315312, 10.04674646, 10.6845459998, 10.849039931, 11.4057042384, 12.0550574465, 12.3293222118, 12.5007858232, 13.1212673911, 13.8393624636, 14.7838884289, 15.2911501434, 15.7684513703, 16.0010833644, 16.3515682839

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