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Properties

Label 2-117-13.12-c15-0-80
Degree $2$
Conductor $117$
Sign $-0.287 - 0.957i$
Analytic cond. $166.951$
Root an. cond. $12.9209$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 141. i·2-s + 1.26e4·4-s − 2.69e5i·5-s − 1.58e6i·7-s − 6.44e6i·8-s − 3.82e7·10-s + 8.17e7i·11-s + (−6.49e7 − 2.16e8i)13-s − 2.25e8·14-s − 5.01e8·16-s + 2.01e9·17-s − 6.31e9i·19-s − 3.39e9i·20-s + 1.15e10·22-s + 2.44e9·23-s + ⋯
L(s)  = 1  − 0.784i·2-s + 0.384·4-s − 1.54i·5-s − 0.729i·7-s − 1.08i·8-s − 1.20·10-s + 1.26i·11-s + (−0.287 − 0.957i)13-s − 0.571·14-s − 0.466·16-s + 1.18·17-s − 1.61i·19-s − 0.593i·20-s + 0.991·22-s + 0.149·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(166.951\)
Root analytic conductor: \(12.9209\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :15/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.876877895\)
\(L(\frac12)\) \(\approx\) \(1.876877895\)
\(L(\frac{17}{2})\) 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 \( 1 + (6.49e7 + 2.16e8i)T \)
good2 \( 1 + 141. iT - 3.27e4T^{2} \)
5 \( 1 + 2.69e5iT - 3.05e10T^{2} \)
7 \( 1 + 1.58e6iT - 4.74e12T^{2} \)
11 \( 1 - 8.17e7iT - 4.17e15T^{2} \)
17 \( 1 - 2.01e9T + 2.86e18T^{2} \)
19 \( 1 + 6.31e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.44e9T + 2.66e20T^{2} \)
29 \( 1 + 1.75e11T + 8.62e21T^{2} \)
31 \( 1 - 2.52e10iT - 2.34e22T^{2} \)
37 \( 1 - 1.06e12iT - 3.33e23T^{2} \)
41 \( 1 + 1.72e12iT - 1.55e24T^{2} \)
43 \( 1 - 6.66e11T + 3.17e24T^{2} \)
47 \( 1 + 5.73e11iT - 1.20e25T^{2} \)
53 \( 1 + 5.50e12T + 7.31e25T^{2} \)
59 \( 1 + 3.69e13iT - 3.65e26T^{2} \)
61 \( 1 + 2.65e13T + 6.02e26T^{2} \)
67 \( 1 + 6.41e13iT - 2.46e27T^{2} \)
71 \( 1 - 6.55e13iT - 5.87e27T^{2} \)
73 \( 1 - 1.34e14iT - 8.90e27T^{2} \)
79 \( 1 - 1.46e14T + 2.91e28T^{2} \)
83 \( 1 + 1.34e14iT - 6.11e28T^{2} \)
89 \( 1 + 1.76e14iT - 1.74e29T^{2} \)
97 \( 1 - 6.00e14iT - 6.33e29T^{2} \)
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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

−9.980907146693840207909668408479, −9.304110686344123413227353222035, −7.85238395513076839655800710963, −7.02242281203174886804356729871, −5.36231982655158483864651704936, −4.49385410728513841594028492817, −3.33143243311357556978332579921, −1.99195234974457053326713476585, −1.06908415916260596002000820567, −0.34478032791600774666583239728, 1.72299131097790058209710697166, 2.74160719881927667780471575694, 3.65564058270148946617546046942, 5.76953711343264145385677767857, 5.99830533824648485530263452846, 7.24699607599001283467610154867, 7.929412395273664807649412662532, 9.267707542378928009648314137141, 10.59421944468685721558675237442, 11.32840814343528962902731064315

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