Properties

Degree 4
Conductor $ 7^{2} \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 − 3·3-s − 2·4-s − 2·5-s + 3·6-s − 7-s + 3·8-s + 3·9-s + 2·10-s + 11-s + 6·12-s − 2·13-s + 14-s + 6·15-s + 16-s + 17-s − 3·18-s − 2·19-s + 4·20-s + 3·21-s − 22-s − 3·23-s − 9·24-s + 25-s + 2·26-s + 2·28-s − 6·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 4-s − 0.894·5-s + 1.22·6-s − 0.377·7-s + 1.06·8-s + 9-s + 0.632·10-s + 0.301·11-s + 1.73·12-s − 0.554·13-s + 0.267·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.458·19-s + 0.894·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s − 1.83·24-s + 1/5·25-s + 0.392·26-s + 0.377·28-s − 1.09·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1519\)    =    \(7^{2} \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1519} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1519,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_2$ \( 1 + T + p T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 11 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + T - 76 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T + 80 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 64 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
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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

−19.2838543194, −18.9180506906, −18.2659420984, −17.9603476504, −17.4457044075, −16.7899227307, −16.637641304, −16.1215262817, −15.2085462244, −14.6748973322, −13.9560995505, −13.0471888346, −12.6916205512, −11.7999974519, −11.6323179436, −10.8078414143, −10.2380344311, −9.4766947638, −8.90147213363, −8.01984494357, −7.33784229568, −6.32627353655, −5.54341957287, −4.7164967317, −3.83210316757, 0, 3.83210316757, 4.7164967317, 5.54341957287, 6.32627353655, 7.33784229568, 8.01984494357, 8.90147213363, 9.4766947638, 10.2380344311, 10.8078414143, 11.6323179436, 11.7999974519, 12.6916205512, 13.0471888346, 13.9560995505, 14.6748973322, 15.2085462244, 16.1215262817, 16.637641304, 16.7899227307, 17.4457044075, 17.9603476504, 18.2659420984, 18.9180506906, 19.2838543194

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