Properties

Degree $4$
Conductor $614656$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·9-s + 14·19-s + 7·25-s − 14·27-s − 12·29-s + 10·31-s − 10·37-s + 6·47-s − 18·53-s + 28·57-s + 18·59-s + 14·75-s − 4·81-s + 24·83-s − 24·87-s + 20·93-s − 2·103-s + 22·109-s − 20·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 12·141-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s + 3.21·19-s + 7/5·25-s − 2.69·27-s − 2.22·29-s + 1.79·31-s − 1.64·37-s + 0.875·47-s − 2.47·53-s + 3.70·57-s + 2.34·59-s + 1.61·75-s − 4/9·81-s + 2.63·83-s − 2.57·87-s + 2.07·93-s − 0.197·103-s + 2.10·109-s − 1.89·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.01·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66796\)
\(L(\frac12)\) \(\approx\) \(2.66796\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 107 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−10.41708011885085121909774219107, −9.860002516453963791533339154698, −9.594033853931277846300165412456, −9.138308019286908159521128785271, −8.845778233884623005822856928439, −8.502887213813762192742809824189, −7.78018782298229755560295735743, −7.74749511872022700380690954945, −7.25451356725258149891540215935, −6.66902675232716157113402250767, −6.07617530251102485833902398952, −5.48587411695623371841732715980, −5.24060217905519585047031327708, −4.79363137965197666229912936064, −3.72130528961797749442251659621, −3.42832494404980997039994527689, −3.06593996374930817045727118253, −2.53217417965741757745191656103, −1.78157633969838083959998193263, −0.78327429415185054960029689807, 0.78327429415185054960029689807, 1.78157633969838083959998193263, 2.53217417965741757745191656103, 3.06593996374930817045727118253, 3.42832494404980997039994527689, 3.72130528961797749442251659621, 4.79363137965197666229912936064, 5.24060217905519585047031327708, 5.48587411695623371841732715980, 6.07617530251102485833902398952, 6.66902675232716157113402250767, 7.25451356725258149891540215935, 7.74749511872022700380690954945, 7.78018782298229755560295735743, 8.502887213813762192742809824189, 8.845778233884623005822856928439, 9.138308019286908159521128785271, 9.594033853931277846300165412456, 9.860002516453963791533339154698, 10.41708011885085121909774219107

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