Properties

Degree 4
Conductor $ 3^{3} \cdot 2221 $
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 − 3·3-s + 4·4-s − 5·5-s + 9·6-s − 5·7-s − 3·8-s + 6·9-s + 15·10-s − 9·11-s − 12·12-s − 2·13-s + 15·14-s + 15·15-s + 3·16-s − 7·17-s − 18·18-s − 6·19-s − 20·20-s + 15·21-s + 27·22-s − 4·23-s + 9·24-s + 11·25-s + 6·26-s − 9·27-s − 20·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 2.23·5-s + 3.67·6-s − 1.88·7-s − 1.06·8-s + 2·9-s + 4.74·10-s − 2.71·11-s − 3.46·12-s − 0.554·13-s + 4.00·14-s + 3.87·15-s + 3/4·16-s − 1.69·17-s − 4.24·18-s − 1.37·19-s − 4.47·20-s + 3.27·21-s + 5.75·22-s − 0.834·23-s + 1.83·24-s + 11/5·25-s + 1.17·26-s − 1.73·27-s − 3.77·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 59967 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 59967 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(59967\)    =    \(3^{3} \cdot 2221\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{59967} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 59967,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;2221\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;2221\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
2221$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 76 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 61 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 112 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 45 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 131 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 188 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−15.6317317498, −15.1512249585, −14.9873535923, −13.4614023964, −13.0943844927, −12.7699549061, −12.403939882, −11.9733672478, −11.2087864077, −11.1093191344, −10.5393790153, −10.1755170206, −9.97755974992, −9.19118150654, −8.68967540446, −8.0868531657, −7.76951160336, −7.28522413861, −6.85417059855, −6.22194294403, −5.63307158998, −4.73282784713, −4.2614919574, −3.37674811938, −2.40345904635, 0, 0, 0, 2.40345904635, 3.37674811938, 4.2614919574, 4.73282784713, 5.63307158998, 6.22194294403, 6.85417059855, 7.28522413861, 7.76951160336, 8.0868531657, 8.68967540446, 9.19118150654, 9.97755974992, 10.1755170206, 10.5393790153, 11.1093191344, 11.2087864077, 11.9733672478, 12.403939882, 12.7699549061, 13.0943844927, 13.4614023964, 14.9873535923, 15.1512249585, 15.6317317498

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