Properties

Degree 4
Conductor $ 2^{7} \cdot 5 \cdot 7^{2} $
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-s + 4-s + 2·5-s − 7-s − 8-s + 9-s − 2·10-s − 3·11-s + 7·13-s + 14-s + 16-s − 18-s + 2·20-s + 3·22-s + 2·25-s − 7·26-s − 28-s + 7·31-s − 32-s − 2·35-s + 36-s − 2·40-s − 8·43-s − 3·44-s + 2·45-s − 6·49-s − 2·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.904·11-s + 1.94·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.447·20-s + 0.639·22-s + 2/5·25-s − 1.37·26-s − 0.188·28-s + 1.25·31-s − 0.176·32-s − 0.338·35-s + 1/6·36-s − 0.316·40-s − 1.21·43-s − 0.452·44-s + 0.298·45-s − 6/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(31360\)    =    \(2^{7} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{31360} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 31360,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.081845150\)
\(L(\frac12)\)  \(\approx\)  \(1.081845150\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−10.24556985624594594311832437286, −10.04717529889385049717314960261, −9.547583722262941142482084390727, −8.867750210995944263052906190176, −8.346613382838925952082229419791, −8.105142939918901360203727371411, −7.21469460361833134925256040423, −6.55198031712863665405316451580, −6.26440470944207612720535785459, −5.57977751846415453797343727473, −4.96653725756449179936568974839, −3.92138491228709829901239666356, −3.17345016701446205336794238050, −2.28291011644118458831469125601, −1.23954203065599301070367986927, 1.23954203065599301070367986927, 2.28291011644118458831469125601, 3.17345016701446205336794238050, 3.92138491228709829901239666356, 4.96653725756449179936568974839, 5.57977751846415453797343727473, 6.26440470944207612720535785459, 6.55198031712863665405316451580, 7.21469460361833134925256040423, 8.105142939918901360203727371411, 8.346613382838925952082229419791, 8.867750210995944263052906190176, 9.547583722262941142482084390727, 10.04717529889385049717314960261, 10.24556985624594594311832437286

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