Properties

Label 2-334620-1.1-c1-0-63
Degree $2$
Conductor $334620$
Sign $1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s + 11-s − 2·19-s + 25-s − 8·31-s + 2·35-s − 2·37-s + 2·43-s − 3·49-s − 6·53-s − 55-s − 12·59-s + 2·61-s + 4·67-s − 2·73-s − 2·77-s − 10·79-s − 12·83-s − 6·89-s + 2·95-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.458·19-s + 1/5·25-s − 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.234·73-s − 0.227·77-s − 1.12·79-s − 1.31·83-s − 0.635·89-s + 0.205·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{334620} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ 1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.88038575394940, −12.57794658013673, −12.31967377673270, −11.60661477688525, −11.20595325930396, −10.88581488730376, −10.30030408043384, −9.816355880147079, −9.412598894174651, −8.941970610400357, −8.501728565841800, −8.011834351963512, −7.401246531769898, −7.068732932989499, −6.592350618027162, −5.994245517381368, −5.712688864875512, −4.940758879633772, −4.523956650459786, −3.894504532478311, −3.513824646503657, −2.990426724096196, −2.408091595615342, −1.670768424338607, −1.146594659229675, 0, 0, 1.146594659229675, 1.670768424338607, 2.408091595615342, 2.990426724096196, 3.513824646503657, 3.894504532478311, 4.523956650459786, 4.940758879633772, 5.712688864875512, 5.994245517381368, 6.592350618027162, 7.068732932989499, 7.401246531769898, 8.011834351963512, 8.501728565841800, 8.941970610400357, 9.412598894174651, 9.816355880147079, 10.30030408043384, 10.88581488730376, 11.20595325930396, 11.60661477688525, 12.31967377673270, 12.57794658013673, 12.88038575394940

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