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Properties

Label 2-76440-1.1-c1-0-73
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 + 2·11-s − 13-s + 15-s + 2·17-s + 6·19-s − 4·23-s + 25-s + 27-s − 8·29-s − 8·31-s + 2·33-s + 2·37-s − 39-s − 6·41-s − 4·43-s + 45-s − 4·47-s + 2·51-s + 6·53-s + 2·55-s + 6·57-s + 12·61-s − 65-s − 6·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.258·15-s + 0.485·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s + 0.280·51-s + 0.824·53-s + 0.269·55-s + 0.794·57-s + 1.53·61-s − 0.124·65-s − 0.733·67-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−14.21682217359437, −13.97325262089367, −13.16662264521022, −13.07765780919213, −12.25006473499423, −11.87833270797704, −11.31872942099752, −10.82822150538654, −9.903236147021132, −9.872210207279867, −9.299649245554518, −8.825064784434629, −8.135103284459144, −7.710192966553673, −7.083164365551524, −6.745506878101509, −5.888699038480473, −5.411576099374495, −5.024464417467117, −3.993866464774796, −3.698330972957374, −3.091672325949277, −2.289443516147209, −1.726725181020124, −1.115909915804026, 0, 1.115909915804026, 1.726725181020124, 2.289443516147209, 3.091672325949277, 3.698330972957374, 3.993866464774796, 5.024464417467117, 5.411576099374495, 5.888699038480473, 6.745506878101509, 7.083164365551524, 7.710192966553673, 8.135103284459144, 8.825064784434629, 9.299649245554518, 9.872210207279867, 9.903236147021132, 10.82822150538654, 11.31872942099752, 11.87833270797704, 12.25006473499423, 13.07765780919213, 13.16662264521022, 13.97325262089367, 14.21682217359437

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