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Properties

Label 2-117-117.61-c1-0-7
Degree $2$
Conductor $117$
Sign $0.179 + 0.983i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0816 + 0.141i)2-s + (−1.72 + 0.158i)3-s + (0.986 − 1.70i)4-s + (−1.55 − 2.69i)5-s + (−0.163 − 0.231i)6-s − 0.136·7-s + 0.649·8-s + (2.94 − 0.547i)9-s + (0.254 − 0.440i)10-s + (−2.08 − 3.61i)11-s + (−1.43 + 3.10i)12-s + (2.96 + 2.05i)13-s + (−0.0111 − 0.0193i)14-s + (3.11 + 4.40i)15-s + (−1.92 − 3.32i)16-s + (2.67 + 4.64i)17-s + ⋯
L(s)  = 1  + (0.0577 + 0.100i)2-s + (−0.995 + 0.0915i)3-s + (0.493 − 0.854i)4-s + (−0.696 − 1.20i)5-s + (−0.0666 − 0.0943i)6-s − 0.0515·7-s + 0.229·8-s + (0.983 − 0.182i)9-s + (0.0804 − 0.139i)10-s + (−0.629 − 1.09i)11-s + (−0.413 + 0.896i)12-s + (0.821 + 0.570i)13-s + (−0.00297 − 0.00515i)14-s + (0.804 + 1.13i)15-s + (−0.480 − 0.831i)16-s + (0.649 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.179 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607158 - 0.506296i\)
\(L(\frac12)\) \(\approx\) \(0.607158 - 0.506296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.158i)T \)
13 \( 1 + (-2.96 - 2.05i)T \)
good2 \( 1 + (-0.0816 - 0.141i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.55 + 2.69i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.136T + 7T^{2} \)
11 \( 1 + (2.08 + 3.61i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.67 - 4.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.154 + 0.268i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.92T + 23T^{2} \)
29 \( 1 + (-1.37 - 2.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.28 - 3.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.48 + 7.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.13T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (0.663 - 1.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.10T + 53T^{2} \)
59 \( 1 + (-2.40 + 4.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + (-2.24 - 3.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.18T + 73T^{2} \)
79 \( 1 + (1.41 - 2.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.19 - 8.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.17 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−13.07959918526499677496165201140, −12.18258814884577002010269470458, −11.17237784637842743821692197147, −10.50727755719036548955643164949, −9.068549948274196239386961924941, −7.81582300921080017962049969121, −6.22635691299309980102879610280, −5.46401818371130586854188358288, −4.17353492837457954055954310640, −1.04917120513311675928351831358, 2.83116344719001614635239796434, 4.32368353486799478372082837463, 6.12211093147143846795077716375, 7.29275172798668453704991198727, 7.79235273612694959000805834559, 9.987086803115274865658122861225, 10.91688817504429182629327956131, 11.65417527525954801124823760222, 12.43908343716967462273571277150, 13.49492710099731755732754247769

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