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Properties

Label 2-117-13.5-c8-0-17
Degree $2$
Conductor $117$
Sign $0.936 - 0.350i$
Analytic cond. $47.6632$
Root an. cond. $6.90386$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.74 − 9.74i)2-s + 65.9i·4-s + (198. − 198. i)5-s + (−1.52e3 − 1.52e3i)7-s + (3.13e3 + 3.13e3i)8-s − 3.87e3i·10-s + (2.96e3 + 2.96e3i)11-s + (1.82e3 + 2.85e4i)13-s − 2.96e4·14-s + 4.42e4·16-s − 3.23e4i·17-s + (−1.06e4 + 1.06e4i)19-s + (1.31e4 + 1.31e4i)20-s + 5.77e4·22-s − 6.04e4i·23-s + ⋯
L(s)  = 1  + (0.609 − 0.609i)2-s + 0.257i·4-s + (0.317 − 0.317i)5-s + (−0.633 − 0.633i)7-s + (0.766 + 0.766i)8-s − 0.387i·10-s + (0.202 + 0.202i)11-s + (0.0639 + 0.997i)13-s − 0.771·14-s + 0.675·16-s − 0.387i·17-s + (−0.0818 + 0.0818i)19-s + (0.0819 + 0.0819i)20-s + 0.246·22-s − 0.216i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(47.6632\)
Root analytic conductor: \(6.90386\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :4),\ 0.936 - 0.350i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.842773431\)
\(L(\frac12)\) \(\approx\) \(2.842773431\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-1.82e3 - 2.85e4i)T \)
good2 \( 1 + (-9.74 + 9.74i)T - 256iT^{2} \)
5 \( 1 + (-198. + 198. i)T - 3.90e5iT^{2} \)
7 \( 1 + (1.52e3 + 1.52e3i)T + 5.76e6iT^{2} \)
11 \( 1 + (-2.96e3 - 2.96e3i)T + 2.14e8iT^{2} \)
17 \( 1 + 3.23e4iT - 6.97e9T^{2} \)
19 \( 1 + (1.06e4 - 1.06e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 6.04e4iT - 7.83e10T^{2} \)
29 \( 1 - 5.49e5T + 5.00e11T^{2} \)
31 \( 1 + (1.94e4 - 1.94e4i)T - 8.52e11iT^{2} \)
37 \( 1 + (-1.93e6 - 1.93e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (2.53e5 - 2.53e5i)T - 7.98e12iT^{2} \)
43 \( 1 - 5.09e6iT - 1.16e13T^{2} \)
47 \( 1 + (-6.02e6 - 6.02e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 1.10e7T + 6.22e13T^{2} \)
59 \( 1 + (1.69e6 + 1.69e6i)T + 1.46e14iT^{2} \)
61 \( 1 - 1.11e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.60e7 + 1.60e7i)T - 4.06e14iT^{2} \)
71 \( 1 + (2.04e7 - 2.04e7i)T - 6.45e14iT^{2} \)
73 \( 1 + (2.81e7 + 2.81e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 6.53e6T + 1.51e15T^{2} \)
83 \( 1 + (-3.82e7 + 3.82e7i)T - 2.25e15iT^{2} \)
89 \( 1 + (-3.40e7 - 3.40e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (1.29e7 - 1.29e7i)T - 7.83e15iT^{2} \)
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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

−12.09042657678840820073332253870, −11.22563357788263613710102719270, −10.03455436336177809045614996843, −8.975886942737026211014681247998, −7.59918842168244942713898679768, −6.44075610318301640935882106756, −4.83026651078843193695043791044, −3.89177800301688616168857265785, −2.63733329750449461241367568262, −1.22987290657104600619493512721, 0.68492145879161777890058205507, 2.45766465846004593220971280874, 3.92322467375532554299083678142, 5.46394254359760408589060037725, 6.10584620687934191677784799196, 7.17595866995885241284314127328, 8.637422298433608603413154862012, 9.922021120463436073128581784377, 10.66042151639703654659911405516, 12.18755222053546856815319764562

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