Properties

Degree 4
Conductor $ 2^{7} \cdot 193 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 3·5-s − 6·7-s + 7·9-s − 13-s + 12·15-s − 9·17-s − 2·19-s + 24·21-s − 6·23-s + 25-s − 4·27-s + 29-s + 2·31-s + 18·35-s + 2·37-s + 4·39-s − 8·43-s − 21·45-s + 15·49-s + 36·51-s + 7·53-s + 8·57-s − 2·59-s + 3·61-s − 42·63-s + 3·65-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.34·5-s − 2.26·7-s + 7/3·9-s − 0.277·13-s + 3.09·15-s − 2.18·17-s − 0.458·19-s + 5.23·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.185·29-s + 0.359·31-s + 3.04·35-s + 0.328·37-s + 0.640·39-s − 1.21·43-s − 3.13·45-s + 15/7·49-s + 5.04·51-s + 0.961·53-s + 1.05·57-s − 0.260·59-s + 0.384·61-s − 5.29·63-s + 0.372·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(24704\)    =    \(2^{7} \cdot 193\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{24704} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 24704,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;193\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;193\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
193$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 17 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
31$D_{4}$ \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \)
59$D_{4}$ \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 66 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T - 53 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−16.1483051938, −15.7835329652, −15.2833487544, −14.9698930036, −13.876726349, −13.3649405453, −12.9328332489, −12.5317509922, −11.9910524239, −11.7091093966, −11.3873667344, −10.7413888829, −10.3721672931, −9.84563669314, −9.23619427149, −8.55134998213, −7.86833193035, −6.89477780439, −6.73115701283, −6.24593319538, −5.8112969284, −4.95432871817, −4.23397773651, −3.76231070724, −2.63730795332, 0, 0, 2.63730795332, 3.76231070724, 4.23397773651, 4.95432871817, 5.8112969284, 6.24593319538, 6.73115701283, 6.89477780439, 7.86833193035, 8.55134998213, 9.23619427149, 9.84563669314, 10.3721672931, 10.7413888829, 11.3873667344, 11.7091093966, 11.9910524239, 12.5317509922, 12.9328332489, 13.3649405453, 13.876726349, 14.9698930036, 15.2833487544, 15.7835329652, 16.1483051938

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