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Properties

Label 2-117600-1.1-c1-0-78
Degree $2$
Conductor $117600$
Sign $1$
Analytic cond. $939.040$
Root an. cond. $30.6437$
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 + 9-s + 7·13-s − 5·17-s + 2·19-s − 3·23-s + 27-s + 3·29-s + 9·31-s − 8·37-s + 7·39-s + 3·41-s − 43-s + 8·47-s − 5·51-s + 3·53-s + 2·57-s − 7·59-s + 61-s + 12·67-s − 3·69-s + 12·71-s + 4·73-s + 12·79-s + 81-s + 3·83-s + 3·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.94·13-s − 1.21·17-s + 0.458·19-s − 0.625·23-s + 0.192·27-s + 0.557·29-s + 1.61·31-s − 1.31·37-s + 1.12·39-s + 0.468·41-s − 0.152·43-s + 1.16·47-s − 0.700·51-s + 0.412·53-s + 0.264·57-s − 0.911·59-s + 0.128·61-s + 1.46·67-s − 0.361·69-s + 1.42·71-s + 0.468·73-s + 1.35·79-s + 1/9·81-s + 0.329·83-s + 0.321·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117600\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(939.040\)
Root analytic conductor: \(30.6437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 117600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.121510351\)
\(L(\frac12)\) \(\approx\) \(4.121510351\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.71274585089395, −13.25878241000224, −12.72650387181565, −12.04340623491132, −11.78315982305186, −10.94434593979653, −10.79330360517912, −10.24334159706692, −9.541710445541876, −9.145291079615780, −8.588348031366698, −8.240115459356473, −7.866697012459356, −7.002968327186336, −6.573920817923067, −6.204308473300796, −5.540259786908986, −4.862816524117426, −4.269580152410172, −3.733796228076845, −3.333893977423164, −2.511523750740255, −2.038255057248996, −1.217580923559131, −0.6494445549681861, 0.6494445549681861, 1.217580923559131, 2.038255057248996, 2.511523750740255, 3.333893977423164, 3.733796228076845, 4.269580152410172, 4.862816524117426, 5.540259786908986, 6.204308473300796, 6.573920817923067, 7.002968327186336, 7.866697012459356, 8.240115459356473, 8.588348031366698, 9.145291079615780, 9.541710445541876, 10.24334159706692, 10.79330360517912, 10.94434593979653, 11.78315982305186, 12.04340623491132, 12.72650387181565, 13.25878241000224, 13.71274585089395

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