Properties

Degree 4
Conductor 53623
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 4-s − 4·5-s + 8·6-s − 7·7-s + 7·9-s + 8·10-s − 2·11-s − 4·12-s − 4·13-s + 14·14-s + 16·15-s + 16-s − 4·17-s − 14·18-s − 4·19-s − 4·20-s + 28·21-s + 4·22-s − 10·23-s + 5·25-s + 8·26-s − 4·27-s − 7·28-s − 32·30-s − 8·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 2.64·7-s + 7/3·9-s + 2.52·10-s − 0.603·11-s − 1.15·12-s − 1.10·13-s + 3.74·14-s + 4.13·15-s + 1/4·16-s − 0.970·17-s − 3.29·18-s − 0.917·19-s − 0.894·20-s + 6.11·21-s + 0.852·22-s − 2.08·23-s + 25-s + 1.56·26-s − 0.769·27-s − 1.32·28-s − 5.84·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53623 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53623 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(53623\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{53623} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 53623,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 53623$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 53623$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad53623$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 48 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$D_{4}$ \( 1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 76 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T + 145 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 148 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 5 T - 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
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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

−15.7373125116, −15.2608168532, −14.8220893029, −13.8734610774, −13.1646945811, −12.6920831477, −12.4931299863, −12.0803253315, −11.6117067570, −11.2712737601, −10.6802560157, −10.2545948213, −9.84164416850, −9.51306438318, −8.86773238113, −8.23916496204, −7.71380458914, −7.12955396154, −6.63792184377, −6.10113065388, −5.86066602875, −4.89933136483, −4.20332644020, −3.64109743427, −2.63589073853, 0, 0, 0, 2.63589073853, 3.64109743427, 4.20332644020, 4.89933136483, 5.86066602875, 6.10113065388, 6.63792184377, 7.12955396154, 7.71380458914, 8.23916496204, 8.86773238113, 9.51306438318, 9.84164416850, 10.2545948213, 10.6802560157, 11.2712737601, 11.6117067570, 12.0803253315, 12.4931299863, 12.6920831477, 13.1646945811, 13.8734610774, 14.8220893029, 15.2608168532, 15.7373125116

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