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Properties

Label 2-117117-1.1-c1-0-31
Degree $2$
Conductor $117117$
Sign $1$
Analytic cond. $935.183$
Root an. cond. $30.5807$
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  + 2-s − 4-s + 3·5-s + 7-s − 3·8-s + 3·10-s + 11-s + 14-s − 16-s + 3·17-s − 3·20-s + 22-s + 8·23-s + 4·25-s − 28-s − 5·29-s + 6·31-s + 5·32-s + 3·34-s + 3·35-s − 3·37-s − 9·40-s − 5·41-s + 4·43-s − 44-s + 8·46-s + 4·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.377·7-s − 1.06·8-s + 0.948·10-s + 0.301·11-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.670·20-s + 0.213·22-s + 1.66·23-s + 4/5·25-s − 0.188·28-s − 0.928·29-s + 1.07·31-s + 0.883·32-s + 0.514·34-s + 0.507·35-s − 0.493·37-s − 1.42·40-s − 0.780·41-s + 0.609·43-s − 0.150·44-s + 1.17·46-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117117\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(935.183\)
Root analytic conductor: \(30.5807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 117117,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.000241244\)
\(L(\frac12)\) \(\approx\) \(5.000241244\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.60580616515767, −13.21835998047029, −12.72046806116518, −12.36377292518603, −11.75995435315805, −11.20632814837425, −10.68808467147720, −10.10268022947016, −9.560095826121170, −9.330961686817587, −8.784169173046763, −8.237984285896266, −7.673652635499173, −6.785043109434552, −6.586606145279745, −5.809720831826693, −5.411937252022870, −5.111990011751190, −4.492971993444397, −3.857423884203489, −3.202072932843095, −2.726966112848150, −1.977798193030195, −1.297706571546160, −0.6445628585619958, 0.6445628585619958, 1.297706571546160, 1.977798193030195, 2.726966112848150, 3.202072932843095, 3.857423884203489, 4.492971993444397, 5.111990011751190, 5.411937252022870, 5.809720831826693, 6.586606145279745, 6.785043109434552, 7.673652635499173, 8.237984285896266, 8.784169173046763, 9.330961686817587, 9.560095826121170, 10.10268022947016, 10.68808467147720, 11.20632814837425, 11.75995435315805, 12.36377292518603, 12.72046806116518, 13.21835998047029, 13.60580616515767

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