Properties

Label 4-966e2-1.1-c1e2-0-7
Degree $4$
Conductor $933156$
Sign $1$
Analytic cond. $59.4988$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s + 2·10-s + 6·12-s + 5·13-s + 4·14-s − 2·15-s + 5·16-s − 6·18-s + 2·19-s − 3·20-s − 4·21-s − 2·23-s − 8·24-s + 25-s − 10·26-s + 4·27-s − 6·28-s + 7·29-s + 4·30-s + 12·31-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 0.632·10-s + 1.73·12-s + 1.38·13-s + 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.41·18-s + 0.458·19-s − 0.670·20-s − 0.872·21-s − 0.417·23-s − 1.63·24-s + 1/5·25-s − 1.96·26-s + 0.769·27-s − 1.13·28-s + 1.29·29-s + 0.730·30-s + 2.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(933156\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(59.4988\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{966} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 933156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.708701608\)
\(L(\frac12)\) \(\approx\) \(1.708701608\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 5 T + 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 218 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−9.983781874018989969865777907399, −9.860581681420956797531759186485, −9.365177163657749865299330200826, −8.820673214720440549220562579229, −8.493622714794371981303560464464, −8.452888141285712836172941424241, −7.75173931181191872057782594951, −7.61262486954545317966996722401, −6.89253921530535890639520588773, −6.70154313407607083989485715057, −6.01478067007850531862752235801, −5.90876413493259073290651989128, −4.86630425022447769465014053937, −4.26420864617901539745847991860, −3.73918192404056483558940036534, −3.29607028429502174270589481137, −2.59454545221232768006274794575, −2.43590585805091429662015644520, −1.23771561526534958407070322239, −0.843104700239064260327536674643, 0.843104700239064260327536674643, 1.23771561526534958407070322239, 2.43590585805091429662015644520, 2.59454545221232768006274794575, 3.29607028429502174270589481137, 3.73918192404056483558940036534, 4.26420864617901539745847991860, 4.86630425022447769465014053937, 5.90876413493259073290651989128, 6.01478067007850531862752235801, 6.70154313407607083989485715057, 6.89253921530535890639520588773, 7.61262486954545317966996722401, 7.75173931181191872057782594951, 8.452888141285712836172941424241, 8.493622714794371981303560464464, 8.820673214720440549220562579229, 9.365177163657749865299330200826, 9.860581681420956797531759186485, 9.983781874018989969865777907399

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