Properties

Degree $4$
Conductor $665856$
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  − 9-s + 4·13-s + 2·17-s + 8·19-s + 10·25-s + 8·43-s + 16·47-s − 2·49-s + 12·53-s + 24·59-s − 24·67-s + 81-s − 24·83-s − 20·89-s − 12·101-s − 4·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.10·13-s + 0.485·17-s + 1.83·19-s + 2·25-s + 1.21·43-s + 2.33·47-s − 2/7·49-s + 1.64·53-s + 3.12·59-s − 2.93·67-s + 1/9·81-s − 2.63·83-s − 2.11·89-s − 1.19·101-s − 0.369·117-s + 6/11·121-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 − 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(665856\)    =    \(2^{8} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{816} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 665856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48419\)
\(L(\frac12)\) \(\approx\) \(2.48419\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−10.36729441438693682621504955593, −10.15236340175770776494369711976, −9.602328290184690979558291126884, −9.062012836245030306475549787344, −8.677379097525106227934651201590, −8.602474759245174437297129837887, −7.86965371605991490770153890036, −7.34587126078453464042934746312, −7.09585719645274874954349186482, −6.68472521839852822760758759176, −5.79428079475028492709609299346, −5.68580132492195408925575053402, −5.33493735024098629651162152676, −4.56656649062571355611528055195, −4.02089302581035142580518405045, −3.58132618596494580605195105364, −2.74483378635255103417511074674, −2.68526237322172352158582426614, −1.29882060825980296171502281625, −0.946357922086703167434109693855, 0.946357922086703167434109693855, 1.29882060825980296171502281625, 2.68526237322172352158582426614, 2.74483378635255103417511074674, 3.58132618596494580605195105364, 4.02089302581035142580518405045, 4.56656649062571355611528055195, 5.33493735024098629651162152676, 5.68580132492195408925575053402, 5.79428079475028492709609299346, 6.68472521839852822760758759176, 7.09585719645274874954349186482, 7.34587126078453464042934746312, 7.86965371605991490770153890036, 8.602474759245174437297129837887, 8.677379097525106227934651201590, 9.062012836245030306475549787344, 9.602328290184690979558291126884, 10.15236340175770776494369711976, 10.36729441438693682621504955593

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