Properties

Degree 4
Conductor $ 2 \cdot 5 \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  − 4-s − 2·5-s + 3·7-s − 2·8-s − 2·9-s − 3·11-s + 16-s + 17-s + 5·19-s + 2·20-s − 5·23-s + 2·25-s − 3·28-s + 4·29-s − 2·31-s + 4·32-s − 6·35-s + 2·36-s + 2·37-s + 4·40-s + 3·41-s + 3·44-s + 4·45-s + 2·47-s − 49-s + 9·53-s + 6·55-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 1.13·7-s − 0.707·8-s − 2/3·9-s − 0.904·11-s + 1/4·16-s + 0.242·17-s + 1.14·19-s + 0.447·20-s − 1.04·23-s + 2/5·25-s − 0.566·28-s + 0.742·29-s − 0.359·31-s + 0.707·32-s − 1.01·35-s + 1/3·36-s + 0.328·37-s + 0.632·40-s + 0.468·41-s + 0.452·44-s + 0.596·45-s + 0.291·47-s − 1/7·49-s + 1.23·53-s + 0.809·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1270 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1270 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1270\)    =    \(2 \cdot 5 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1270} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1270,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4736509421$
$L(\frac12)$  $\approx$  $0.4736509421$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;127\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;127\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
127$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 20 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$D_{4}$ \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$D_{4}$ \( 1 - 5 T + 26 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$V_4$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$V_4$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−19.6568311241, −18.9503890316, −18.3576321607, −17.9939273287, −17.6312819661, −16.8834354892, −16.0779288654, −15.7672744879, −14.9597293671, −14.606176504, −13.9435487587, −13.4462724835, −12.4378997441, −11.9874616952, −11.4861866002, −10.8390478871, −10.0245475085, −9.20293686385, −8.32131772067, −8.04671447301, −7.28147039781, −5.89944013519, −5.2055600604, −4.23659828419, −2.97079164477, 2.97079164477, 4.23659828419, 5.2055600604, 5.89944013519, 7.28147039781, 8.04671447301, 8.32131772067, 9.20293686385, 10.0245475085, 10.8390478871, 11.4861866002, 11.9874616952, 12.4378997441, 13.4462724835, 13.9435487587, 14.606176504, 14.9597293671, 15.7672744879, 16.0779288654, 16.8834354892, 17.6312819661, 17.9939273287, 18.3576321607, 18.9503890316, 19.6568311241

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