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Properties

Label 2-117600-1.1-c1-0-169
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 4·11-s − 6·13-s + 6·17-s − 4·19-s − 4·23-s + 27-s − 2·29-s + 8·31-s + 4·33-s − 6·37-s − 6·39-s − 6·41-s + 8·43-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s − 10·61-s + 8·67-s − 4·69-s − 12·71-s − 14·73-s + 16·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 1.21·43-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.977·67-s − 0.481·69-s − 1.42·71-s − 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-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: \(1\)
Selberg data: \((2,\ 117600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.06499144178908, −13.45786017725805, −12.75600097515275, −12.22473064871721, −12.05810305294785, −11.67988082176298, −10.71546907892499, −10.38307106858261, −9.814288742705464, −9.473899112565849, −9.028677632984692, −8.300914787668901, −7.988885851750805, −7.360443468659168, −6.996481376356118, −6.306604465866965, −5.888667256297956, −5.139560285240066, −4.514421346305765, −4.195823976357878, −3.357577190675514, −3.033508871734043, −2.171360028333785, −1.744593795024749, −0.9376168547004779, 0, 0.9376168547004779, 1.744593795024749, 2.171360028333785, 3.033508871734043, 3.357577190675514, 4.195823976357878, 4.514421346305765, 5.139560285240066, 5.888667256297956, 6.306604465866965, 6.996481376356118, 7.360443468659168, 7.988885851750805, 8.300914787668901, 9.028677632984692, 9.473899112565849, 9.814288742705464, 10.38307106858261, 10.71546907892499, 11.67988082176298, 12.05810305294785, 12.22473064871721, 12.75600097515275, 13.45786017725805, 14.06499144178908

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