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Properties

Label 2-117-13.3-c3-0-8
Degree $2$
Conductor $117$
Sign $-0.787 + 0.616i$
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  + (−2.69 − 4.67i)2-s + (−10.5 + 18.2i)4-s + 13.0·5-s + (3.21 − 5.56i)7-s + 70.7·8-s + (−35.1 − 60.9i)10-s + (−13.1 − 22.6i)11-s + (43.2 − 18.0i)13-s − 34.6·14-s + (−106. − 184. i)16-s + (61.9 − 107. i)17-s + (−54.8 + 94.9i)19-s + (−137. + 238. i)20-s + (−70.7 + 122. i)22-s + (31.7 + 54.9i)23-s + ⋯
L(s)  = 1  + (−0.953 − 1.65i)2-s + (−1.31 + 2.28i)4-s + 1.16·5-s + (0.173 − 0.300i)7-s + 3.12·8-s + (−1.11 − 1.92i)10-s + (−0.359 − 0.622i)11-s + (0.922 − 0.385i)13-s − 0.661·14-s + (−1.66 − 2.88i)16-s + (0.883 − 1.53i)17-s + (−0.662 + 1.14i)19-s + (−1.53 + 2.66i)20-s + (−0.685 + 1.18i)22-s + (0.287 + 0.497i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.787 + 0.616i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.787 + 0.616i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.355093 - 1.02961i\)
\(L(\frac12)\) \(\approx\) \(0.355093 - 1.02961i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-43.2 + 18.0i)T \)
good2 \( 1 + (2.69 + 4.67i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 13.0T + 125T^{2} \)
7 \( 1 + (-3.21 + 5.56i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (13.1 + 22.6i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-61.9 + 107. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (54.8 - 94.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-31.7 - 54.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (112. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 200.T + 2.97e4T^{2} \)
37 \( 1 + (126. + 218. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (113. + 196. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-192. + 332. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 34.6T + 1.03e5T^{2} \)
53 \( 1 + 61.0T + 1.48e5T^{2} \)
59 \( 1 + (-40.2 + 69.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-13.0 + 22.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-465. - 806. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (213. - 370. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 108.T + 3.89e5T^{2} \)
79 \( 1 - 384.T + 4.93e5T^{2} \)
83 \( 1 + 85.9T + 5.71e5T^{2} \)
89 \( 1 + (-247. - 429. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (95.4 - 165. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.40735686636302669837565874746, −11.35801871675525214414705402884, −10.42603011253301720587731252733, −9.765832827968220253091256449723, −8.749821559711573559549610632532, −7.66946784717096624909357088155, −5.61277249301705047986390396323, −3.70872406506193018582380429583, −2.33624065763265132797319037002, −0.903282662158869739723650622829, 1.55909587933586545232878417688, 4.86452292140865323135960799878, 5.98990720596768966658576794283, 6.70951707281526525200390978400, 8.143485195532710886605508764387, 8.952658374283234334786857376754, 9.935386651266312234124055024437, 10.76121783767532237219824301035, 12.90686379099114738374210611721, 13.80436084501512128578883758843

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