Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s − 2·11-s − 9·13-s + 2·19-s − 5·23-s − 5·25-s + 15·29-s + 31-s + 2·35-s + 3·37-s − 19·41-s − 2·43-s + 3·47-s + 3·49-s − 53-s − 2·55-s + 8·59-s − 9·65-s − 19·67-s + 71-s − 6·73-s − 4·77-s − 13·79-s − 83-s − 15·89-s − 18·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 0.603·11-s − 2.49·13-s + 0.458·19-s − 1.04·23-s − 25-s + 2.78·29-s + 0.179·31-s + 0.338·35-s + 0.493·37-s − 2.96·41-s − 0.304·43-s + 0.437·47-s + 3/7·49-s − 0.137·53-s − 0.269·55-s + 1.04·59-s − 1.11·65-s − 2.32·67-s + 0.118·71-s − 0.702·73-s − 0.455·77-s − 1.46·79-s − 0.109·83-s − 1.58·89-s − 1.88·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 42 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 15 T + 110 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 19 T + 168 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 19 T + 186 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 13 T + 196 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T + 162 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.966386697539174720372448275490, −7.63292790909831278025212069472, −7.05327003267082296599275194264, −7.00986558144595677201286053493, −6.55390925413442249522444153366, −6.12274594274799119789308011504, −5.59577654618211323280145063643, −5.37074666119075625883512125789, −4.98479823992853945645033545571, −4.77635569414471297549996406512, −4.19002059562842036588146228474, −4.16723405678429839411115725631, −3.21062348569067196964645614477, −2.92881154578593966218939079090, −2.39000379768590684286381703799, −2.34409633930843878821108498046, −1.51787588069036585793746864117, −1.27012914838403455209423860209, 0, 0, 1.27012914838403455209423860209, 1.51787588069036585793746864117, 2.34409633930843878821108498046, 2.39000379768590684286381703799, 2.92881154578593966218939079090, 3.21062348569067196964645614477, 4.16723405678429839411115725631, 4.19002059562842036588146228474, 4.77635569414471297549996406512, 4.98479823992853945645033545571, 5.37074666119075625883512125789, 5.59577654618211323280145063643, 6.12274594274799119789308011504, 6.55390925413442249522444153366, 7.00986558144595677201286053493, 7.05327003267082296599275194264, 7.63292790909831278025212069472, 7.966386697539174720372448275490

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