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Properties

Label 2-117-13.4-c3-0-2
Degree $2$
Conductor $117$
Sign $0.227 - 0.973i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (−2.5 − 4.33i)4-s + 1.73i·5-s + (−12 + 6.92i)7-s + 22.5i·8-s + (1.49 − 2.59i)10-s + (−12 − 6.92i)11-s + (45.5 + 11.2i)13-s + 24·14-s + (−0.500 + 0.866i)16-s + (58.5 + 101. i)17-s + (−99 + 57.1i)19-s + (7.49 − 4.33i)20-s + (12 + 20.7i)22-s + (−39 + 67.5i)23-s + ⋯
L(s)  = 1  + (−0.530 − 0.306i)2-s + (−0.312 − 0.541i)4-s + 0.154i·5-s + (−0.647 + 0.374i)7-s + 0.995i·8-s + (0.0474 − 0.0821i)10-s + (−0.328 − 0.189i)11-s + (0.970 + 0.240i)13-s + 0.458·14-s + (−0.00781 + 0.0135i)16-s + (0.834 + 1.44i)17-s + (−1.19 + 0.690i)19-s + (0.0838 − 0.0484i)20-s + (0.116 + 0.201i)22-s + (−0.353 + 0.612i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.227 - 0.973i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.227 - 0.973i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.502435 + 0.398488i\)
\(L(\frac12)\) \(\approx\) \(0.502435 + 0.398488i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-45.5 - 11.2i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 1.73iT - 125T^{2} \)
7 \( 1 + (12 - 6.92i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (12 + 6.92i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-58.5 - 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (99 - 57.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (39 - 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (70.5 - 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + (124.5 + 71.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (235.5 + 135. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (52 + 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 301. iT - 1.03e5T^{2} \)
53 \( 1 + 93T + 1.48e5T^{2} \)
59 \( 1 + (-246 + 142. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (681 + 393. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (915 - 528. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 458. iT - 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 - 789. iT - 5.71e5T^{2} \)
89 \( 1 + (-846 - 488. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-174 + 100. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.20717528902327138288559179387, −12.22521295036010144953983761294, −10.76225884822375867593847032029, −10.30991335696103781081526290736, −8.991786613071356610945517737969, −8.267858943814520333303857331635, −6.43722383010280546816219765876, −5.46227900219245376534047894049, −3.59766407506016470266247373318, −1.64447917202708820271422566668, 0.42098454798999874767220131192, 3.13455995609953690567489845437, 4.55750188168561727517801399791, 6.35833433875422507205743815823, 7.45343929781162028545063759085, 8.516998791206060649721819729527, 9.486748251421329213926989668180, 10.49570431799921544407060974096, 11.88863427732678576769302060962, 13.03924393388091608205364278754

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