Properties

Label 4-6762e2-1.1-c1e2-0-11
Degree $4$
Conductor $45724644$
Sign $1$
Analytic cond. $2915.44$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 4·8-s + 3·9-s − 8·10-s − 6·12-s + 8·15-s + 5·16-s + 6·18-s + 4·19-s − 12·20-s − 2·23-s − 8·24-s + 2·25-s − 4·27-s − 4·29-s + 16·30-s − 4·31-s + 6·32-s + 9·36-s − 4·37-s + 8·38-s − 16·40-s + 12·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 1.41·8-s + 9-s − 2.52·10-s − 1.73·12-s + 2.06·15-s + 5/4·16-s + 1.41·18-s + 0.917·19-s − 2.68·20-s − 0.417·23-s − 1.63·24-s + 2/5·25-s − 0.769·27-s − 0.742·29-s + 2.92·30-s − 0.718·31-s + 1.06·32-s + 3/2·36-s − 0.657·37-s + 1.29·38-s − 2.52·40-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(45724644\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2915.44\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6762} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 45724644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.56949522208829419460220157969, −7.46240187672602840361923130008, −7.12384169358857544224533234657, −6.68561576903558979322592974762, −6.23024237043829594777686184890, −6.00086531050659593905619193889, −5.55498928191957357784180483372, −5.38527935029534650080003363422, −4.90097811643310037092845697735, −4.51389790047837359627473796013, −4.18153104469561100531937631122, −3.94143220501659481170953589650, −3.43642960235211622329161557337, −3.43440665376120551085979052233, −2.45568744021517076318742775169, −2.39710679754217439154470131666, −1.29491499148409965249544355272, −1.26458621099679501445844531273, 0, 0, 1.26458621099679501445844531273, 1.29491499148409965249544355272, 2.39710679754217439154470131666, 2.45568744021517076318742775169, 3.43440665376120551085979052233, 3.43642960235211622329161557337, 3.94143220501659481170953589650, 4.18153104469561100531937631122, 4.51389790047837359627473796013, 4.90097811643310037092845697735, 5.38527935029534650080003363422, 5.55498928191957357784180483372, 6.00086531050659593905619193889, 6.23024237043829594777686184890, 6.68561576903558979322592974762, 7.12384169358857544224533234657, 7.46240187672602840361923130008, 7.56949522208829419460220157969

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