Properties

Degree 4
Conductor $ 2 \cdot 3491 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4-s − 5·5-s + 6·6-s − 4·7-s + 2·8-s + 2·9-s + 10·10-s − 3·11-s − 3·12-s + 8·14-s + 15·15-s − 3·16-s − 2·17-s − 4·18-s − 19-s − 5·20-s + 12·21-s + 6·22-s + 23-s − 6·24-s + 10·25-s + 6·27-s − 4·28-s − 6·29-s − 30·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.23·5-s + 2.44·6-s − 1.51·7-s + 0.707·8-s + 2/3·9-s + 3.16·10-s − 0.904·11-s − 0.866·12-s + 2.13·14-s + 3.87·15-s − 3/4·16-s − 0.485·17-s − 0.942·18-s − 0.229·19-s − 1.11·20-s + 2.61·21-s + 1.27·22-s + 0.208·23-s − 1.22·24-s + 2·25-s + 1.15·27-s − 0.755·28-s − 1.11·29-s − 5.47·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6982 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6982 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6982\)    =    \(2 \cdot 3491\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6982} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 6982,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3491\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3491\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
3491$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 92 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 13 T + 95 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T - 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 7 T + 98 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 11 T + 113 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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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

−17.5070115827, −16.9668742634, −16.6010923109, −16.2929666273, −15.8439924964, −15.5040319431, −14.9096557708, −14.0397581253, −13.0382123975, −12.7194278991, −12.3450096294, −11.5210091973, −11.1747224381, −10.9764198772, −10.2571832692, −9.63478157041, −8.98131598229, −8.35444068068, −7.70117114998, −7.3120607735, −6.54500289869, −5.79558712082, −4.99611129295, −4.0726099633, −3.23822879075, 0, 0, 3.23822879075, 4.0726099633, 4.99611129295, 5.79558712082, 6.54500289869, 7.3120607735, 7.70117114998, 8.35444068068, 8.98131598229, 9.63478157041, 10.2571832692, 10.9764198772, 11.1747224381, 11.5210091973, 12.3450096294, 12.7194278991, 13.0382123975, 14.0397581253, 14.9096557708, 15.5040319431, 15.8439924964, 16.2929666273, 16.6010923109, 16.9668742634, 17.5070115827

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