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Properties

Label 2-102960-1.1-c1-0-12
Degree $2$
Conductor $102960$
Sign $1$
Analytic cond. $822.139$
Root an. cond. $28.6729$
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  + 5-s − 7-s + 11-s − 13-s − 4·17-s + 3·19-s − 6·23-s + 25-s − 4·29-s + 7·31-s − 35-s − 8·37-s + 6·41-s + 5·43-s + 9·47-s − 6·49-s + 4·53-s + 55-s + 5·59-s + 13·61-s − 65-s − 11·67-s − 16·71-s + 4·73-s − 77-s + 79-s − 13·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.277·13-s − 0.970·17-s + 0.688·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s + 1.25·31-s − 0.169·35-s − 1.31·37-s + 0.937·41-s + 0.762·43-s + 1.31·47-s − 6/7·49-s + 0.549·53-s + 0.134·55-s + 0.650·59-s + 1.66·61-s − 0.124·65-s − 1.34·67-s − 1.89·71-s + 0.468·73-s − 0.113·77-s + 0.112·79-s − 1.42·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(822.139\)
Root analytic conductor: \(28.6729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 102960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867579891\)
\(L(\frac12)\) \(\approx\) \(1.867579891\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good7 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.72496165742047, −13.30008876747730, −12.76756178640969, −12.29852027937473, −11.67042712899584, −11.48805882397855, −10.62542382988706, −10.28079753883654, −9.806893739707423, −9.276867131664209, −8.846785691916027, −8.341699720321175, −7.639734070727533, −7.153850872865113, −6.691746779962792, −6.008841219888272, −5.730609631050403, −5.047385885584356, −4.322940472919253, −3.974199496054530, −3.199634075927488, −2.538671646801956, −2.054279247249351, −1.283243152308397, −0.4320136842483331, 0.4320136842483331, 1.283243152308397, 2.054279247249351, 2.538671646801956, 3.199634075927488, 3.974199496054530, 4.322940472919253, 5.047385885584356, 5.730609631050403, 6.008841219888272, 6.691746779962792, 7.153850872865113, 7.639734070727533, 8.341699720321175, 8.846785691916027, 9.276867131664209, 9.806893739707423, 10.28079753883654, 10.62542382988706, 11.48805882397855, 11.67042712899584, 12.29852027937473, 12.76756178640969, 13.30008876747730, 13.72496165742047

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