Properties

Degree $4$
Conductor $1285956$
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  + 2·2-s + 3·4-s + 2·5-s − 2·7-s + 4·8-s + 4·10-s + 4·11-s − 4·14-s + 5·16-s + 4·17-s + 10·19-s + 6·20-s + 8·22-s − 2·23-s − 25-s − 6·28-s − 4·29-s + 12·31-s + 6·32-s + 8·34-s − 4·35-s + 4·37-s + 20·38-s + 8·40-s + 4·43-s + 12·44-s − 4·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s + 1.41·8-s + 1.26·10-s + 1.20·11-s − 1.06·14-s + 5/4·16-s + 0.970·17-s + 2.29·19-s + 1.34·20-s + 1.70·22-s − 0.417·23-s − 1/5·25-s − 1.13·28-s − 0.742·29-s + 2.15·31-s + 1.06·32-s + 1.37·34-s − 0.676·35-s + 0.657·37-s + 3.24·38-s + 1.26·40-s + 0.609·43-s + 1.80·44-s − 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.821428039\)
\(L(\frac12)\) \(\approx\) \(7.821428039\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−9.871789683529587753535228693256, −9.793494786784438501371574931727, −9.351794020520797996072876028654, −9.046057312394664916492155775652, −8.097411820302655090278509220683, −7.944597885090500187252565392832, −7.37329037389303157850569248409, −6.93646771189539582000617549097, −6.46185032101993303340984275912, −6.16482514106445268229372327869, −5.79017344906158182702940994675, −5.34585629002613168293784273338, −5.00531107582065291770556216251, −4.30194463606659594268058444029, −3.81694937894669710576354089081, −3.47190081021483249752507638496, −2.78007502426693272047252555679, −2.57532945852329164575102546344, −1.45426245522405647146436173250, −1.15623785622867158010014689522, 1.15623785622867158010014689522, 1.45426245522405647146436173250, 2.57532945852329164575102546344, 2.78007502426693272047252555679, 3.47190081021483249752507638496, 3.81694937894669710576354089081, 4.30194463606659594268058444029, 5.00531107582065291770556216251, 5.34585629002613168293784273338, 5.79017344906158182702940994675, 6.16482514106445268229372327869, 6.46185032101993303340984275912, 6.93646771189539582000617549097, 7.37329037389303157850569248409, 7.944597885090500187252565392832, 8.097411820302655090278509220683, 9.046057312394664916492155775652, 9.351794020520797996072876028654, 9.793494786784438501371574931727, 9.871789683529587753535228693256

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