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Properties

Label 2-117-13.8-c4-0-2
Degree $2$
Conductor $117$
Sign $0.621 - 0.783i$
Analytic cond. $12.0942$
Root an. cond. $3.47768$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.89 − 2.89i)2-s + 0.794i·4-s + (−18.6 − 18.6i)5-s + (−15.2 + 15.2i)7-s + (−44.0 + 44.0i)8-s + 108. i·10-s + (59.2 − 59.2i)11-s + (−157. − 62.1i)13-s + 88.4·14-s + 268.·16-s + 272. i·17-s + (−54.5 − 54.5i)19-s + (14.7 − 14.7i)20-s − 343.·22-s + 596. i·23-s + ⋯
L(s)  = 1  + (−0.724 − 0.724i)2-s + 0.0496i·4-s + (−0.745 − 0.745i)5-s + (−0.311 + 0.311i)7-s + (−0.688 + 0.688i)8-s + 1.08i·10-s + (0.489 − 0.489i)11-s + (−0.929 − 0.367i)13-s + 0.451·14-s + 1.04·16-s + 0.944i·17-s + (−0.151 − 0.151i)19-s + (0.0369 − 0.0369i)20-s − 0.709·22-s + 1.12i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(12.0942\)
Root analytic conductor: \(3.47768\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :2),\ 0.621 - 0.783i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.273837 + 0.132308i\)
\(L(\frac12)\) \(\approx\) \(0.273837 + 0.132308i\)
\(L(3)\) 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 \( 1 + (157. + 62.1i)T \)
good2 \( 1 + (2.89 + 2.89i)T + 16iT^{2} \)
5 \( 1 + (18.6 + 18.6i)T + 625iT^{2} \)
7 \( 1 + (15.2 - 15.2i)T - 2.40e3iT^{2} \)
11 \( 1 + (-59.2 + 59.2i)T - 1.46e4iT^{2} \)
17 \( 1 - 272. iT - 8.35e4T^{2} \)
19 \( 1 + (54.5 + 54.5i)T + 1.30e5iT^{2} \)
23 \( 1 - 596. iT - 2.79e5T^{2} \)
29 \( 1 - 1.31e3T + 7.07e5T^{2} \)
31 \( 1 + (-874. - 874. i)T + 9.23e5iT^{2} \)
37 \( 1 + (403. - 403. i)T - 1.87e6iT^{2} \)
41 \( 1 + (983. + 983. i)T + 2.82e6iT^{2} \)
43 \( 1 - 2.23e3iT - 3.41e6T^{2} \)
47 \( 1 + (-369. + 369. i)T - 4.87e6iT^{2} \)
53 \( 1 + 4.39e3T + 7.89e6T^{2} \)
59 \( 1 + (976. - 976. i)T - 1.21e7iT^{2} \)
61 \( 1 + 6.16e3T + 1.38e7T^{2} \)
67 \( 1 + (1.70e3 + 1.70e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (672. + 672. i)T + 2.54e7iT^{2} \)
73 \( 1 + (1.16e3 - 1.16e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 8.34e3T + 3.89e7T^{2} \)
83 \( 1 + (6.51e3 + 6.51e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (8.36e3 - 8.36e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (-2.27e3 - 2.27e3i)T + 8.85e7iT^{2} \)
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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

−12.40492568065488529007770781916, −11.99574603236569803394132812963, −10.81860724230724128350732723239, −9.796923032212019006198409841490, −8.799499670783226980878832100955, −7.999302132826739259419196081051, −6.21705818836592392628344529609, −4.78543016240351124701585684555, −3.05972325402831586382193188075, −1.24338494762895988061052666745, 0.18599498228617313607110483223, 2.95149523723459462041548071638, 4.41761641986629402111492464332, 6.56626704810458446913640860368, 7.13873604112219604032810034547, 8.126936010947787542754820798201, 9.373173424475014187048822310643, 10.29594065338023389053078112020, 11.73279247397586424109152530156, 12.42874543869023398224102854683

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