Properties

Degree 4
Conductor $ 3 \cdot 13331 $
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 − 6·5-s + 9·6-s − 6·7-s − 3·8-s + 4·9-s + 18·10-s − 4·11-s − 12·12-s − 5·13-s + 18·14-s + 18·15-s + 3·16-s − 4·17-s − 12·18-s − 5·19-s − 24·20-s + 18·21-s + 12·22-s − 8·23-s + 9·24-s + 18·25-s + 15·26-s − 24·28-s − 29-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 4/3·9-s + 5.69·10-s − 1.20·11-s − 3.46·12-s − 1.38·13-s + 4.81·14-s + 4.64·15-s + 3/4·16-s − 0.970·17-s − 2.82·18-s − 1.14·19-s − 5.36·20-s + 3.92·21-s + 2.55·22-s − 1.66·23-s + 1.83·24-s + 18/5·25-s + 2.94·26-s − 4.53·28-s − 0.185·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 39993 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 39993 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(39993\)    =    \(3 \cdot 13331\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{39993} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 39993,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13331\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;13331\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
13331$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 70 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T - 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T - 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 72 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T - 60 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 109 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T - 47 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 17 T + 163 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
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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

−15.9704645837, −15.6693054009, −15.20560946, −14.7641403995, −13.6456655209, −12.8592480806, −12.5953132339, −12.3023007929, −11.7010824178, −11.5251462003, −10.8563797264, −10.3700922436, −10.0207472259, −9.79274056879, −8.89574331067, −8.38228702161, −8.08029649442, −7.37677982037, −7.08567270792, −6.4894769244, −5.97422717765, −4.98342638789, −4.31718290768, −3.59646164255, −2.70281332604, 0, 0, 0, 2.70281332604, 3.59646164255, 4.31718290768, 4.98342638789, 5.97422717765, 6.4894769244, 7.08567270792, 7.37677982037, 8.08029649442, 8.38228702161, 8.89574331067, 9.79274056879, 10.0207472259, 10.3700922436, 10.8563797264, 11.5251462003, 11.7010824178, 12.3023007929, 12.5953132339, 12.8592480806, 13.6456655209, 14.7641403995, 15.20560946, 15.6693054009, 15.9704645837

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