Properties

Degree 4
Conductor $ 2 \cdot 3 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s − 3·5-s + 7-s + 2·8-s − 3·11-s + 12-s − 2·13-s + 3·15-s + 16-s + 17-s + 5·19-s + 3·20-s − 21-s + 23-s − 2·24-s + 5·25-s + 4·27-s − 28-s − 4·32-s + 3·33-s − 3·35-s − 4·37-s + 2·39-s − 6·40-s − 9·41-s − 6·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.707·8-s − 0.904·11-s + 0.288·12-s − 0.554·13-s + 0.774·15-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.670·20-s − 0.218·21-s + 0.208·23-s − 0.408·24-s + 25-s + 0.769·27-s − 0.188·28-s − 0.707·32-s + 0.522·33-s − 0.507·35-s − 0.657·37-s + 0.320·39-s − 0.948·40-s − 1.40·41-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(762\)    =    \(2 \cdot 3 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{762} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 762,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3486135246$
$L(\frac12)$  $\approx$  $0.3486135246$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;127\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;127\}$, 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$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
127$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 16 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \)
97$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−19.9054817718, −19.325537269, −18.7707813896, −18.2292872568, −17.7135368063, −16.9790062377, −16.418397115, −15.9571764584, −15.3067109583, −14.6962325998, −13.9597388923, −13.3185622928, −12.5600821651, −11.8923236171, −11.4770698286, −10.5989964585, −10.1481082508, −9.0333799854, −8.05536019859, −7.69205500554, −6.81393831996, −5.20477015259, −4.83911118217, −3.48602202783, 3.48602202783, 4.83911118217, 5.20477015259, 6.81393831996, 7.69205500554, 8.05536019859, 9.0333799854, 10.1481082508, 10.5989964585, 11.4770698286, 11.8923236171, 12.5600821651, 13.3185622928, 13.9597388923, 14.6962325998, 15.3067109583, 15.9571764584, 16.418397115, 16.9790062377, 17.7135368063, 18.2292872568, 18.7707813896, 19.325537269, 19.9054817718

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