Properties

Degree 4
Conductor 3391
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 − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 2·7-s − 3·8-s + 7·9-s + 15·10-s − 4·11-s − 16·12-s − 2·13-s + 6·14-s + 20·15-s + 3·16-s − 4·17-s − 21·18-s + 2·19-s − 20·20-s + 8·21-s + 12·22-s + 12·24-s + 10·25-s + 6·26-s − 4·27-s − 8·28-s − 2·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 0.755·7-s − 1.06·8-s + 7/3·9-s + 4.74·10-s − 1.20·11-s − 4.61·12-s − 0.554·13-s + 1.60·14-s + 5.16·15-s + 3/4·16-s − 0.970·17-s − 4.94·18-s + 0.458·19-s − 4.47·20-s + 1.74·21-s + 2.55·22-s + 2.44·24-s + 2·25-s + 1.17·26-s − 0.769·27-s − 1.51·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3391 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3391 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3391\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3391} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 3391,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3391$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 3391$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3391$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 80 T + p T^{2} ) \)
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$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 52 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$D_{4}$ \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 94 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 3 T - 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 96 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−18.4591181114, −17.9636956494, −17.6360205721, −17.1089016924, −16.5986459687, −16.2677920166, −15.8858719243, −15.3939816337, −14.9662294560, −13.5512906784, −12.7583925178, −12.1772653521, −11.8078640393, −11.3709692996, −10.8618791166, −10.3158196872, −9.93867452538, −8.86453502441, −8.43021839996, −7.60176719027, −7.24964560736, −6.51129535267, −5.47447461279, −4.79816559228, −3.50770790118, 0, 0, 3.50770790118, 4.79816559228, 5.47447461279, 6.51129535267, 7.24964560736, 7.60176719027, 8.43021839996, 8.86453502441, 9.93867452538, 10.3158196872, 10.8618791166, 11.3709692996, 11.8078640393, 12.1772653521, 12.7583925178, 13.5512906784, 14.9662294560, 15.3939816337, 15.8858719243, 16.2677920166, 16.5986459687, 17.1089016924, 17.6360205721, 17.9636956494, 18.4591181114

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