Properties

Degree $4$
Conductor $4624$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s − 8·13-s + 6·17-s − 8·19-s + 2·25-s + 16·43-s − 24·47-s − 4·49-s − 12·53-s − 8·67-s + 7·81-s + 24·89-s + 24·101-s + 16·103-s − 32·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 4/3·9-s − 2.21·13-s + 1.45·17-s − 1.83·19-s + 2/5·25-s + 2.43·43-s − 3.50·47-s − 4/7·49-s − 1.64·53-s − 0.977·67-s + 7/9·81-s + 2.54·89-s + 2.38·101-s + 1.57·103-s − 2.95·117-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8285555715\)
\(L(\frac12)\) \(\approx\) \(0.8285555715\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−12.42812820711201065307788952263, −11.96651015706018560929811900846, −11.09976168376602818722871050760, −10.45018406564663794713022036780, −9.859947069031234632947769625138, −9.668885494194043070932761404458, −8.775324792952608305234038274451, −7.73194344000881192909461523211, −7.60550254480635953750503585943, −6.73888792302150141204335430044, −6.06047072166510840925822598058, −4.78095038515256975486634747518, −4.64030908745122342322785253550, −3.31155621017059602303168202463, −2.02390448372635546016211511373, 2.02390448372635546016211511373, 3.31155621017059602303168202463, 4.64030908745122342322785253550, 4.78095038515256975486634747518, 6.06047072166510840925822598058, 6.73888792302150141204335430044, 7.60550254480635953750503585943, 7.73194344000881192909461523211, 8.775324792952608305234038274451, 9.668885494194043070932761404458, 9.859947069031234632947769625138, 10.45018406564663794713022036780, 11.09976168376602818722871050760, 11.96651015706018560929811900846, 12.42812820711201065307788952263

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