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Properties

Label 2-117-13.10-c7-0-18
Degree $2$
Conductor $117$
Sign $0.919 - 0.393i$
Analytic cond. $36.5490$
Root an. cond. $6.04558$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.0 − 6.37i)2-s + (17.3 − 30.1i)4-s + 28.6i·5-s + (−344. − 199. i)7-s + 1.18e3i·8-s + (182. + 316. i)10-s + (4.40e3 − 2.54e3i)11-s + (−3.21e3 − 7.24e3i)13-s − 5.08e3·14-s + (9.81e3 + 1.69e4i)16-s + (−1.93e4 + 3.34e4i)17-s + (4.46e4 + 2.57e4i)19-s + (861. + 497. i)20-s + (3.24e4 − 5.62e4i)22-s + (2.78e4 + 4.83e4i)23-s + ⋯
L(s)  = 1  + (0.976 − 0.563i)2-s + (0.135 − 0.235i)4-s + 0.102i·5-s + (−0.380 − 0.219i)7-s + 0.821i·8-s + (0.0577 + 0.0999i)10-s + (0.998 − 0.576i)11-s + (−0.405 − 0.914i)13-s − 0.494·14-s + (0.598 + 1.03i)16-s + (−0.953 + 1.65i)17-s + (1.49 + 0.861i)19-s + (0.0240 + 0.0139i)20-s + (0.650 − 1.12i)22-s + (0.478 + 0.828i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(36.5490\)
Root analytic conductor: \(6.04558\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :7/2),\ 0.919 - 0.393i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.144417721\)
\(L(\frac12)\) \(\approx\) \(3.144417721\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.21e3 + 7.24e3i)T \)
good2 \( 1 + (-11.0 + 6.37i)T + (64 - 110. i)T^{2} \)
5 \( 1 - 28.6iT - 7.81e4T^{2} \)
7 \( 1 + (344. + 199. i)T + (4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-4.40e3 + 2.54e3i)T + (9.74e6 - 1.68e7i)T^{2} \)
17 \( 1 + (1.93e4 - 3.34e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-4.46e4 - 2.57e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-2.78e4 - 4.83e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-6.41e4 - 1.11e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 - 2.50e4iT - 2.75e10T^{2} \)
37 \( 1 + (-1.19e4 + 6.92e3i)T + (4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (-3.91e5 + 2.25e5i)T + (9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (6.99e4 - 1.21e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 - 1.17e6iT - 5.06e11T^{2} \)
53 \( 1 + 8.40e5T + 1.17e12T^{2} \)
59 \( 1 + (-8.92e5 - 5.14e5i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.24e6 + 2.16e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.83e6 - 1.06e6i)T + (3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-2.35e6 - 1.36e6i)T + (4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 3.33e6iT - 1.10e13T^{2} \)
79 \( 1 - 9.81e5T + 1.92e13T^{2} \)
83 \( 1 + 1.38e6iT - 2.71e13T^{2} \)
89 \( 1 + (-3.93e6 + 2.27e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (5.24e5 + 3.02e5i)T + (4.03e13 + 6.99e13i)T^{2} \)
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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

−12.44841429529827371454465776542, −11.36043236563598108744595434329, −10.48700367861232795275802769864, −9.083020094834090472995705372116, −7.898287810778108952705168540997, −6.37728936504804750826757472905, −5.23382062821378801680165630208, −3.83825243329709575830081064388, −3.05619662321406706944459076276, −1.33073128864326739463464438607, 0.74454875073318150411503696909, 2.74954335052537444920294350424, 4.34033566980953875057234001409, 5.08135723427152180977811209584, 6.63829358632699870237260783315, 7.08057331967407439862266119186, 9.139632220883884375571478608212, 9.662454482579440054520611615722, 11.44812171534153104340284132171, 12.24732527270451207285089560228

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