Properties

Degree 4
Conductor 59107
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 4·5-s + 12·6-s − 3·7-s − 3·8-s + 8·9-s + 12·10-s − 7·11-s − 16·12-s − 7·13-s + 9·14-s + 16·15-s + 3·16-s − 2·17-s − 24·18-s − 6·19-s − 16·20-s + 12·21-s + 21·22-s − 5·23-s + 12·24-s + 3·25-s + 21·26-s − 12·27-s − 12·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 1.78·5-s + 4.89·6-s − 1.13·7-s − 1.06·8-s + 8/3·9-s + 3.79·10-s − 2.11·11-s − 4.61·12-s − 1.94·13-s + 2.40·14-s + 4.13·15-s + 3/4·16-s − 0.485·17-s − 5.65·18-s − 1.37·19-s − 3.57·20-s + 2.61·21-s + 4.47·22-s − 1.04·23-s + 2.44·24-s + 3/5·25-s + 4.11·26-s − 2.30·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 59107 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 59107 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(59107\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{59107} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 59107,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 59107$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 59107$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad59107$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 200 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 67 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 83 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 11 T + 113 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 15 T + 162 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 9 T + 96 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 20 T + 212 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−15.5450057951, −15.2293167683, −14.8303802617, −13.7478167727, −13.026499971, −12.5947437537, −12.3291008821, −11.9580477307, −11.4672423498, −10.9648611024, −10.6330675053, −10.1414423148, −9.97118241748, −9.41322544, −8.63774794728, −8.0147189586, −7.78954447721, −7.31514554991, −6.80719639994, −6.10208501977, −5.6281809622, −4.82683357676, −4.4578803929, −3.3692209215, −2.26245636956, 0, 0, 0, 2.26245636956, 3.3692209215, 4.4578803929, 4.82683357676, 5.6281809622, 6.10208501977, 6.80719639994, 7.31514554991, 7.78954447721, 8.0147189586, 8.63774794728, 9.41322544, 9.97118241748, 10.1414423148, 10.6330675053, 10.9648611024, 11.4672423498, 11.9580477307, 12.3291008821, 12.5947437537, 13.026499971, 13.7478167727, 14.8303802617, 15.2293167683, 15.5450057951

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