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Properties

Label 2-117-13.12-c11-0-37
Degree $2$
Conductor $117$
Sign $0.842 - 0.537i$
Analytic cond. $89.8961$
Root an. cond. $9.48135$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.2i·2-s + 1.40e3·4-s − 3.30e3i·5-s + 5.10e4i·7-s + 8.74e4i·8-s + 8.35e4·10-s − 3.90e5i·11-s + (1.12e6 − 7.20e5i)13-s − 1.29e6·14-s + 6.72e5·16-s + 6.18e6·17-s − 1.15e7i·19-s − 4.64e6i·20-s + 9.87e6·22-s − 8.42e6·23-s + ⋯
L(s)  = 1  + 0.558i·2-s + 0.687·4-s − 0.472i·5-s + 1.14i·7-s + 0.943i·8-s + 0.264·10-s − 0.730i·11-s + (0.842 − 0.537i)13-s − 0.641·14-s + 0.160·16-s + 1.05·17-s − 1.06i·19-s − 0.324i·20-s + 0.408·22-s − 0.272·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.537i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.842 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.842 - 0.537i$
Analytic conductor: \(89.8961\)
Root analytic conductor: \(9.48135\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :11/2),\ 0.842 - 0.537i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.081218183\)
\(L(\frac12)\) \(\approx\) \(3.081218183\)
\(L(\frac{13}{2})\) 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 \( 1 + (-1.12e6 + 7.20e5i)T \)
good2 \( 1 - 25.2iT - 2.04e3T^{2} \)
5 \( 1 + 3.30e3iT - 4.88e7T^{2} \)
7 \( 1 - 5.10e4iT - 1.97e9T^{2} \)
11 \( 1 + 3.90e5iT - 2.85e11T^{2} \)
17 \( 1 - 6.18e6T + 3.42e13T^{2} \)
19 \( 1 + 1.15e7iT - 1.16e14T^{2} \)
23 \( 1 + 8.42e6T + 9.52e14T^{2} \)
29 \( 1 + 1.57e8T + 1.22e16T^{2} \)
31 \( 1 - 7.17e7iT - 2.54e16T^{2} \)
37 \( 1 - 2.91e7iT - 1.77e17T^{2} \)
41 \( 1 + 9.25e8iT - 5.50e17T^{2} \)
43 \( 1 - 2.68e8T + 9.29e17T^{2} \)
47 \( 1 + 1.75e9iT - 2.47e18T^{2} \)
53 \( 1 - 5.81e9T + 9.26e18T^{2} \)
59 \( 1 - 8.17e8iT - 3.01e19T^{2} \)
61 \( 1 + 6.68e8T + 4.35e19T^{2} \)
67 \( 1 - 1.11e10iT - 1.22e20T^{2} \)
71 \( 1 + 1.62e10iT - 2.31e20T^{2} \)
73 \( 1 - 2.14e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.83e10T + 7.47e20T^{2} \)
83 \( 1 + 2.57e9iT - 1.28e21T^{2} \)
89 \( 1 + 9.33e9iT - 2.77e21T^{2} \)
97 \( 1 - 5.74e10iT - 7.15e21T^{2} \)
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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

−11.54018969709263382167800083439, −10.58297601182823039102363302511, −8.977235469184190311639339555046, −8.354781313948874887951242246591, −7.12457980319554179479411351978, −5.81015402354428535701084803918, −5.35929570263624328866155540146, −3.38114673507208015292786920147, −2.23630549161008262001997369723, −0.845031130732693362681701521144, 0.953646232355366916823013326040, 1.86958578387262666188833979382, 3.30923372157921439343254845350, 4.12897639908539341354803084105, 5.98252363246887113944989598225, 7.04582024680185130794887254199, 7.80395697912270184905741579415, 9.615344735406426952521805927202, 10.42571858351894458523192912243, 11.14899205211636509059619976622

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