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Properties

Label 2-76440-1.1-c1-0-74
Degree $2$
Conductor $76440$
Sign $-1$
Analytic cond. $610.376$
Root an. cond. $24.7057$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 5·11-s + 13-s + 15-s − 17-s − 6·19-s + 3·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 5·33-s − 11·37-s + 39-s − 9·41-s + 6·43-s + 45-s + 4·47-s − 51-s − 6·53-s + 5·55-s − 6·57-s + 9·59-s − 10·61-s + 65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 1.37·19-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.870·33-s − 1.80·37-s + 0.160·39-s − 1.40·41-s + 0.914·43-s + 0.149·45-s + 0.583·47-s − 0.140·51-s − 0.824·53-s + 0.674·55-s − 0.794·57-s + 1.17·59-s − 1.28·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76440\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(610.376\)
Root analytic conductor: \(24.7057\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76440,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.41959908923061, −13.79151724426097, −13.36107408893642, −12.90055479186357, −12.29294870594083, −11.95446323747086, −11.19346326883963, −10.72068177573908, −10.39883662683347, −9.504159231970395, −9.200823684289649, −8.862965791341667, −8.338980315353907, −7.678162491386771, −6.931506097205287, −6.672745440010991, −6.144645697828624, −5.402603763028238, −4.857254413861096, −4.026015542390457, −3.750647719768979, −3.121875008399235, −2.150934443380569, −1.820389386540337, −1.117645805215141, 0, 1.117645805215141, 1.820389386540337, 2.150934443380569, 3.121875008399235, 3.750647719768979, 4.026015542390457, 4.857254413861096, 5.402603763028238, 6.144645697828624, 6.672745440010991, 6.931506097205287, 7.678162491386771, 8.338980315353907, 8.862965791341667, 9.200823684289649, 9.504159231970395, 10.39883662683347, 10.72068177573908, 11.19346326883963, 11.95446323747086, 12.29294870594083, 12.90055479186357, 13.36107408893642, 13.79151724426097, 14.41959908923061

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