Properties

Degree $4$
Conductor $921600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 9-s − 4·11-s + 11·25-s − 16·31-s + 4·41-s − 4·45-s + 10·49-s − 16·55-s + 20·59-s − 4·61-s + 24·71-s + 81-s + 20·89-s + 4·99-s + 16·101-s + 20·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 1.20·11-s + 11/5·25-s − 2.87·31-s + 0.624·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 2.60·59-s − 0.512·61-s + 2.84·71-s + 1/9·81-s + 2.11·89-s + 0.402·99-s + 1.59·101-s + 1.91·109-s − 0.909·121-s + 2.14·125-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 − 5.14·155-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47642\)
\(L(\frac12)\) \(\approx\) \(2.47642\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.12096110965081070063231957866, −10.00753669308419175057550088676, −9.212099851975808375776571483064, −9.193411808798071220306016277263, −8.715158220990760030641822257168, −8.254969611591929950200229108632, −7.48323700282678448355579266359, −7.46062972812106738853262156018, −6.78755993818593172158689370167, −6.27774383454896964527410242383, −5.88131568020044699660890229835, −5.37964832598748357281358339416, −5.24833189380140533966738785239, −4.77142167373993736883932077938, −3.77205334529209662907691509218, −3.50680404998244847127583535008, −2.43558900516617993360117622150, −2.41364340619497992275745377862, −1.76623382470639652667722352824, −0.72102067387008897618786772945, 0.72102067387008897618786772945, 1.76623382470639652667722352824, 2.41364340619497992275745377862, 2.43558900516617993360117622150, 3.50680404998244847127583535008, 3.77205334529209662907691509218, 4.77142167373993736883932077938, 5.24833189380140533966738785239, 5.37964832598748357281358339416, 5.88131568020044699660890229835, 6.27774383454896964527410242383, 6.78755993818593172158689370167, 7.46062972812106738853262156018, 7.48323700282678448355579266359, 8.254969611591929950200229108632, 8.715158220990760030641822257168, 9.193411808798071220306016277263, 9.212099851975808375776571483064, 10.00753669308419175057550088676, 10.12096110965081070063231957866

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