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Properties

Label 2-117-13.12-c11-0-12
Degree $2$
Conductor $117$
Sign $-0.767 - 0.640i$
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  + 38.6i·2-s + 557.·4-s − 1.19e3i·5-s − 7.31e4i·7-s + 1.00e5i·8-s + 4.61e4·10-s + 9.54e5i·11-s + (−1.02e6 − 8.57e5i)13-s + 2.82e6·14-s − 2.74e6·16-s + 2.94e6·17-s − 1.05e7i·19-s − 6.66e5i·20-s − 3.68e7·22-s − 5.56e7·23-s + ⋯
L(s)  = 1  + 0.853i·2-s + 0.272·4-s − 0.171i·5-s − 1.64i·7-s + 1.08i·8-s + 0.145·10-s + 1.78i·11-s + (−0.767 − 0.640i)13-s + 1.40·14-s − 0.653·16-s + 0.503·17-s − 0.973i·19-s − 0.0465i·20-s − 1.52·22-s − 1.80·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.767 - 0.640i$
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: \((2,\ 117,\ (\ :11/2),\ -0.767 - 0.640i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.666668760\)
\(L(\frac12)\) \(\approx\) \(1.666668760\)
\(L(\frac{13}{2})\) 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 \( 1 + (1.02e6 + 8.57e5i)T \)
good2 \( 1 - 38.6iT - 2.04e3T^{2} \)
5 \( 1 + 1.19e3iT - 4.88e7T^{2} \)
7 \( 1 + 7.31e4iT - 1.97e9T^{2} \)
11 \( 1 - 9.54e5iT - 2.85e11T^{2} \)
17 \( 1 - 2.94e6T + 3.42e13T^{2} \)
19 \( 1 + 1.05e7iT - 1.16e14T^{2} \)
23 \( 1 + 5.56e7T + 9.52e14T^{2} \)
29 \( 1 - 4.50e7T + 1.22e16T^{2} \)
31 \( 1 - 1.42e8iT - 2.54e16T^{2} \)
37 \( 1 - 4.72e8iT - 1.77e17T^{2} \)
41 \( 1 - 7.78e7iT - 5.50e17T^{2} \)
43 \( 1 - 4.31e8T + 9.29e17T^{2} \)
47 \( 1 - 5.63e8iT - 2.47e18T^{2} \)
53 \( 1 - 1.65e9T + 9.26e18T^{2} \)
59 \( 1 - 3.87e9iT - 3.01e19T^{2} \)
61 \( 1 - 1.04e10T + 4.35e19T^{2} \)
67 \( 1 - 1.38e10iT - 1.22e20T^{2} \)
71 \( 1 - 1.13e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.01e10iT - 3.13e20T^{2} \)
79 \( 1 + 2.77e9T + 7.47e20T^{2} \)
83 \( 1 - 3.88e10iT - 1.28e21T^{2} \)
89 \( 1 + 2.56e9iT - 2.77e21T^{2} \)
97 \( 1 + 3.20e10iT - 7.15e21T^{2} \)
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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

−11.86295214885197512446067895266, −10.48552677058233981276780929438, −9.889321288055783629671602348070, −8.158302267975757396405477643036, −7.26645501571653859288253444941, −6.79062543389791144518758473202, −5.15705995828826095722683137234, −4.26444781742044559785345327060, −2.54298006778806384172325895673, −1.14850040294695335526913071017, 0.36622718670370748872945230711, 1.88488241107028212805289019012, 2.69945130670519249618511330867, 3.74662312781147715071557506644, 5.60063240976095008744898088742, 6.33127569174308316444067321543, 7.967409324457600416941791142130, 9.039940927826273264883637336925, 10.06513147530960422659669695099, 11.19790282064456484328378820536

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