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Properties

Label 2-117117-1.1-c1-0-28
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 + 4·5-s − 7-s − 3·8-s + 4·10-s − 11-s − 14-s − 16-s + 3·17-s + 8·19-s − 4·20-s − 22-s − 6·23-s + 11·25-s + 28-s + 6·29-s + 4·31-s + 5·32-s + 3·34-s − 4·35-s − 2·37-s + 8·38-s − 12·40-s − 7·41-s + 7·43-s + 44-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.377·7-s − 1.06·8-s + 1.26·10-s − 0.301·11-s − 0.267·14-s − 1/4·16-s + 0.727·17-s + 1.83·19-s − 0.894·20-s − 0.213·22-s − 1.25·23-s + 11/5·25-s + 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.883·32-s + 0.514·34-s − 0.676·35-s − 0.328·37-s + 1.29·38-s − 1.89·40-s − 1.09·41-s + 1.06·43-s + 0.150·44-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

\(L(1)\) \(\approx\) \(4.673712104\)
\(L(\frac12)\) \(\approx\) \(4.673712104\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.67626225071863, −13.43356073410505, −12.57317299443655, −12.34956877927137, −11.96162564035576, −11.20411910030583, −10.48599389744089, −9.965515602962961, −9.755288320575470, −9.407203441040365, −8.809816973442757, −8.146093644865707, −7.742400487015971, −6.714941105409954, −6.536744753884493, −5.803411011517847, −5.497361403768083, −5.142393789996215, −4.529519178949052, −3.800222637658632, −2.979313016478613, −2.921680753348996, −2.013882537897960, −1.290193089301432, −0.6163974653731075, 0.6163974653731075, 1.290193089301432, 2.013882537897960, 2.921680753348996, 2.979313016478613, 3.800222637658632, 4.529519178949052, 5.142393789996215, 5.497361403768083, 5.803411011517847, 6.536744753884493, 6.714941105409954, 7.742400487015971, 8.146093644865707, 8.809816973442757, 9.407203441040365, 9.755288320575470, 9.965515602962961, 10.48599389744089, 11.20411910030583, 11.96162564035576, 12.34956877927137, 12.57317299443655, 13.43356073410505, 13.67626225071863

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