Properties

Degree 4
Conductor $ 2 \cdot 7 \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 2·5-s − 7-s + 2·8-s + 2·9-s + 2·12-s + 4·13-s + 4·15-s + 16-s − 4·17-s − 2·19-s + 2·20-s + 2·21-s − 4·23-s − 4·24-s + 2·25-s − 6·27-s + 28-s + 6·29-s + 4·31-s − 4·32-s + 2·35-s − 2·36-s + 4·37-s − 8·39-s − 4·40-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.707·8-s + 2/3·9-s + 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.447·20-s + 0.436·21-s − 0.834·23-s − 0.816·24-s + 2/5·25-s − 1.15·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.707·32-s + 0.338·35-s − 1/3·36-s + 0.657·37-s − 1.28·39-s − 0.632·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(574\)    =    \(2 \cdot 7 \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{574} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 574,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2886587543\)
\(L(\frac12)\)  \(\approx\)  \(0.2886587543\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;41\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T - 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−19.6733132147, −19.2894810758, −18.7091307297, −17.8708865577, −17.6960182246, −16.8518514821, −16.2735421554, −15.8505407017, −15.3180145253, −14.3426984432, −13.5290037260, −13.0723731511, −12.3156957452, −11.5725833897, −11.0601290334, −10.4541609560, −9.56623783527, −8.50715996695, −7.85008095901, −6.70727794107, −5.99154001731, −4.71800513845, −3.93159440622, 3.93159440622, 4.71800513845, 5.99154001731, 6.70727794107, 7.85008095901, 8.50715996695, 9.56623783527, 10.4541609560, 11.0601290334, 11.5725833897, 12.3156957452, 13.0723731511, 13.5290037260, 14.3426984432, 15.3180145253, 15.8505407017, 16.2735421554, 16.8518514821, 17.6960182246, 17.8708865577, 18.7091307297, 19.2894810758, 19.6733132147

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