Properties

Degree 4
Conductor $ 5 \cdot 1433 $
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 − 4·3-s + 4-s − 3·5-s + 8·6-s − 7-s + 6·9-s + 6·10-s − 6·11-s − 4·12-s − 3·13-s + 2·14-s + 12·15-s + 16-s − 4·17-s − 12·18-s − 19-s − 3·20-s + 4·21-s + 12·22-s − 7·23-s + 8·25-s + 6·26-s + 4·27-s − 28-s + 6·29-s − 24·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 3.26·6-s − 0.377·7-s + 2·9-s + 1.89·10-s − 1.80·11-s − 1.15·12-s − 0.832·13-s + 0.534·14-s + 3.09·15-s + 1/4·16-s − 0.970·17-s − 2.82·18-s − 0.229·19-s − 0.670·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s + 8/5·25-s + 1.17·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 4.38·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7165\)    =    \(5 \cdot 1433\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7165} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 7165,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;1433\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;1433\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
1433$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 34 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 26 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 15 T + 240 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−17.5954652855, −17.1618300644, −16.6734506598, −16.1586599489, −15.7856712111, −15.605000504, −14.7632187118, −13.9930474232, −13.1268514144, −12.5458112695, −12.1198832533, −11.7551998036, −11.2403384152, −10.5464085999, −10.3961483092, −9.88170401658, −8.7586175334, −8.43271993934, −7.82179147315, −7.04015264766, −6.49987538805, −5.67579823181, −5.00604408141, −4.47379472706, −2.92391090258, 0, 0, 2.92391090258, 4.47379472706, 5.00604408141, 5.67579823181, 6.49987538805, 7.04015264766, 7.82179147315, 8.43271993934, 8.7586175334, 9.88170401658, 10.3961483092, 10.5464085999, 11.2403384152, 11.7551998036, 12.1198832533, 12.5458112695, 13.1268514144, 13.9930474232, 14.7632187118, 15.605000504, 15.7856712111, 16.1586599489, 16.6734506598, 17.1618300644, 17.5954652855

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