Properties

Degree $4$
Conductor $4624$
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 + 3·8-s + 2·9-s + 4·13-s − 16-s + 2·17-s − 2·18-s − 6·25-s − 4·26-s − 5·32-s − 2·34-s − 2·36-s − 8·43-s − 8·47-s + 2·49-s + 6·50-s − 4·52-s − 4·53-s − 8·59-s + 7·64-s + 16·67-s − 2·68-s + 6·72-s − 5·81-s − 8·83-s + 8·86-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.10·13-s − 1/4·16-s + 0.485·17-s − 0.471·18-s − 6/5·25-s − 0.784·26-s − 0.883·32-s − 0.342·34-s − 1/3·36-s − 1.21·43-s − 1.16·47-s + 2/7·49-s + 0.848·50-s − 0.554·52-s − 0.549·53-s − 1.04·59-s + 7/8·64-s + 1.95·67-s − 0.242·68-s + 0.707·72-s − 5/9·81-s − 0.878·83-s + 0.862·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4624\)    =    \(2^{4} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4624} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5599681098\)
\(L(\frac12)\) \(\approx\) \(0.5599681098\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + p T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.49705356412274795654430513469, −11.52441822743087419852023092159, −11.16009675289169332696240221176, −10.37457762625256391453072139208, −9.865147796700457696764531203637, −9.503201408039874570785920496525, −8.641089803848343655534817241388, −8.206562194159880247099856720914, −7.60192191739855759685095936356, −6.82967169758881859503844382698, −5.99963448119736311949210049496, −5.13040230457225388037561720767, −4.24397414242561147496719054255, −3.47108995746223846434545530649, −1.59166732572850702666348542306, 1.59166732572850702666348542306, 3.47108995746223846434545530649, 4.24397414242561147496719054255, 5.13040230457225388037561720767, 5.99963448119736311949210049496, 6.82967169758881859503844382698, 7.60192191739855759685095936356, 8.206562194159880247099856720914, 8.641089803848343655534817241388, 9.503201408039874570785920496525, 9.865147796700457696764531203637, 10.37457762625256391453072139208, 11.16009675289169332696240221176, 11.52441822743087419852023092159, 12.49705356412274795654430513469

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