Properties

Degree 4
Conductor $ 7 \cdot 14303 $
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-s − 2·3-s − 2·4-s − 3·5-s + 2·6-s + 2·7-s + 3·8-s − 9-s + 3·10-s − 3·11-s + 4·12-s − 2·14-s + 6·15-s + 16-s − 3·17-s + 18-s + 6·20-s − 4·21-s + 3·22-s + 2·23-s − 6·24-s − 2·25-s + 6·27-s − 4·28-s + 7·29-s − 6·30-s − 10·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s − 1.34·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s − 1/3·9-s + 0.948·10-s − 0.904·11-s + 1.15·12-s − 0.534·14-s + 1.54·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.34·20-s − 0.872·21-s + 0.639·22-s + 0.417·23-s − 1.22·24-s − 2/5·25-s + 1.15·27-s − 0.755·28-s + 1.29·29-s − 1.09·30-s − 1.79·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100121\)    =    \(7 \cdot 14303\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100121} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 100121,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;14303\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;14303\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - T + p T^{2} ) \)
14303$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 204 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$V_4$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 13 T + 109 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 5 T + 64 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$V_4$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 51 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 144 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 13 T + 107 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 15 T + 169 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−14.4844339726, −13.9794895095, −13.666248259, −13.1217124481, −12.6231912595, −12.1634952427, −11.7066397208, −11.3708040639, −11.0378278066, −10.5617615251, −10.157155395, −9.55422340584, −8.84804560685, −8.53302196564, −8.30439290125, −7.63609362078, −7.33493759989, −6.57685380223, −5.89211230329, −5.22890251571, −4.92409084078, −4.46188966538, −3.73599546234, −2.99633556726, −1.64484697581, 0, 0, 1.64484697581, 2.99633556726, 3.73599546234, 4.46188966538, 4.92409084078, 5.22890251571, 5.89211230329, 6.57685380223, 7.33493759989, 7.63609362078, 8.30439290125, 8.53302196564, 8.84804560685, 9.55422340584, 10.157155395, 10.5617615251, 11.0378278066, 11.3708040639, 11.7066397208, 12.1634952427, 12.6231912595, 13.1217124481, 13.666248259, 13.9794895095, 14.4844339726

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