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Properties

Label 2-76440-1.1-c1-0-36
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 4·11-s + 13-s − 15-s + 2·17-s − 2·19-s + 2·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 4·33-s − 2·37-s + 39-s + 10·41-s − 4·43-s − 45-s + 8·47-s + 2·51-s − 4·55-s − 2·57-s + 12·59-s − 8·61-s − 65-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.328·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s − 0.539·55-s − 0.264·57-s + 1.56·59-s − 1.02·61-s − 0.124·65-s + 0.240·69-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: \(0\)
Selberg data: \((2,\ 76440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.953846749\)
\(L(\frac12)\) \(\approx\) \(3.953846749\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.19411314172964, −13.64284186106161, −13.05625866464497, −12.51379870792075, −12.05659920011356, −11.67038268264319, −11.03639936024802, −10.51811898624040, −10.02507574886756, −9.362905936903137, −8.937785957345230, −8.546261331560091, −7.938447410147127, −7.455313810052661, −6.835422465007868, −6.387678975742734, −5.835244176254880, −4.990792799834268, −4.451502354033640, −3.878763857896509, −3.451765843946896, −2.737445471864283, −2.092289820615949, −1.203565654391484, −0.7129616151128372, 0.7129616151128372, 1.203565654391484, 2.092289820615949, 2.737445471864283, 3.451765843946896, 3.878763857896509, 4.451502354033640, 4.990792799834268, 5.835244176254880, 6.387678975742734, 6.835422465007868, 7.455313810052661, 7.938447410147127, 8.546261331560091, 8.937785957345230, 9.362905936903137, 10.02507574886756, 10.51811898624040, 11.03639936024802, 11.67038268264319, 12.05659920011356, 12.51379870792075, 13.05625866464497, 13.64284186106161, 14.19411314172964

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