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Properties

Label 2-117-9.2-c2-0-11
Degree $2$
Conductor $117$
Sign $0.613 - 0.789i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.19 + 1.84i)2-s + (−2.62 − 1.45i)3-s + (4.78 + 8.28i)4-s + (7.14 − 4.12i)5-s + (−5.70 − 9.46i)6-s + (−2.39 + 4.15i)7-s + 20.5i·8-s + (4.78 + 7.62i)9-s + 30.4·10-s + (−12.2 − 7.06i)11-s + (−0.536 − 28.7i)12-s + (1.80 + 3.12i)13-s + (−15.2 + 8.83i)14-s + (−24.7 + 0.462i)15-s + (−18.6 + 32.2i)16-s − 19.3i·17-s + ⋯
L(s)  = 1  + (1.59 + 0.920i)2-s + (−0.875 − 0.483i)3-s + (1.19 + 2.07i)4-s + (1.42 − 0.825i)5-s + (−0.950 − 1.57i)6-s + (−0.342 + 0.593i)7-s + 2.56i·8-s + (0.532 + 0.846i)9-s + 3.04·10-s + (−1.11 − 0.641i)11-s + (−0.0447 − 2.39i)12-s + (0.138 + 0.240i)13-s + (−1.09 + 0.630i)14-s + (−1.65 + 0.0308i)15-s + (−1.16 + 2.01i)16-s − 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.613 - 0.789i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.47701 + 1.21191i\)
\(L(\frac12)\) \(\approx\) \(2.47701 + 1.21191i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.62 + 1.45i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good2 \( 1 + (-3.19 - 1.84i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-7.14 + 4.12i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (2.39 - 4.15i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (12.2 + 7.06i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 + 19.3iT - 289T^{2} \)
19 \( 1 + 12.4T + 361T^{2} \)
23 \( 1 + (6.83 - 3.94i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.567 + 0.327i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.46 + 2.53i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 65.5T + 1.36e3T^{2} \)
41 \( 1 + (-24.7 + 14.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-7.83 + 13.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-59.3 - 34.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 54.1iT - 2.80e3T^{2} \)
59 \( 1 + (41.3 - 23.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.5 - 70.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-26.6 - 46.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 41.0iT - 5.04e3T^{2} \)
73 \( 1 + 12.7T + 5.32e3T^{2} \)
79 \( 1 + (-67.1 + 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-86.4 - 49.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 10.4iT - 7.92e3T^{2} \)
97 \( 1 + (-70.6 + 122. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.57790956701977958987035684570, −12.67644871701404400035993345010, −12.06707090581338709092370427163, −10.61858002471280347548332636818, −8.922431874358970871432335333713, −7.43201530665177185615547818468, −6.11212658846009120777546521445, −5.63498821929614450671684935079, −4.78672274858676219363269240243, −2.46779780707619275535103911354, 2.07283935319548868561533993179, 3.62498583981416343557106342786, 5.03276309343149442699892645269, 5.96425788826780249619441822238, 6.76994141582970310781312134700, 9.836180272944491290950938796104, 10.52590251007965626741888230188, 10.79795523484610385892186232628, 12.39366848634645656215029709311, 13.03762254269407731657641650160

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