Properties

Degree 4
Conductor $ 2^{2} \cdot 13 \cdot 31 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s − 5-s + 4·6-s − 4·7-s + 8-s + 7·9-s + 10-s − 11-s − 2·13-s + 4·14-s + 4·15-s − 16-s + 2·17-s − 7·18-s + 16·21-s + 22-s + 23-s − 4·24-s + 2·25-s + 2·26-s − 4·27-s − 2·29-s − 4·30-s − 3·31-s + 4·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 0.447·5-s + 1.63·6-s − 1.51·7-s + 0.353·8-s + 7/3·9-s + 0.316·10-s − 0.301·11-s − 0.554·13-s + 1.06·14-s + 1.03·15-s − 1/4·16-s + 0.485·17-s − 1.64·18-s + 3.49·21-s + 0.213·22-s + 0.208·23-s − 0.816·24-s + 2/5·25-s + 0.392·26-s − 0.769·27-s − 0.371·29-s − 0.730·30-s − 0.538·31-s + 0.696·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1612 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1612 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1612\)    =    \(2^{2} \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1612} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1612,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T - 19 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T - 17 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T - 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
73$D_{4}$ \( 1 + 3 T + 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T - 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 9 T + 138 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T - 109 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−19.1451606556, −18.9001986874, −18.2593436156, −17.7853307843, −17.1244416671, −16.8861189244, −16.4500779792, −15.9680891095, −15.4934478559, −14.6265579485, −13.6947773651, −12.8699406387, −12.5263891588, −11.9002889637, −11.4403837587, −10.6830957099, −10.2643435694, −9.68064483916, −8.85979049704, −7.83552410185, −6.79655649118, −6.57754311776, −5.46618957048, −5.04047037067, −3.50975460109, 0, 3.50975460109, 5.04047037067, 5.46618957048, 6.57754311776, 6.79655649118, 7.83552410185, 8.85979049704, 9.68064483916, 10.2643435694, 10.6830957099, 11.4403837587, 11.9002889637, 12.5263891588, 12.8699406387, 13.6947773651, 14.6265579485, 15.4934478559, 15.9680891095, 16.4500779792, 16.8861189244, 17.1244416671, 17.7853307843, 18.2593436156, 18.9001986874, 19.1451606556

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