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Properties

Label 2-117-39.5-c1-0-3
Degree $2$
Conductor $117$
Sign $0.347 - 0.937i$
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  + (1.47 + 1.47i)2-s + 2.35i·4-s + (0.955 + 0.955i)5-s + (−3.08 − 3.08i)7-s + (−0.519 + 0.519i)8-s + 2.82i·10-s + (−0.955 + 0.955i)11-s + (−3.43 + 1.08i)13-s − 9.10i·14-s + 3.17·16-s + 4.24·17-s + (2.43 − 2.43i)19-s + (−2.24 + 2.24i)20-s − 2.82·22-s − 7.81·23-s + ⋯
L(s)  = 1  + (1.04 + 1.04i)2-s + 1.17i·4-s + (0.427 + 0.427i)5-s + (−1.16 − 1.16i)7-s + (−0.183 + 0.183i)8-s + 0.891i·10-s + (−0.288 + 0.288i)11-s + (−0.953 + 0.301i)13-s − 2.43i·14-s + 0.793·16-s + 1.02·17-s + (0.559 − 0.559i)19-s + (−0.502 + 0.502i)20-s − 0.601·22-s − 1.62·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34235 + 0.933680i\)
\(L(\frac12)\) \(\approx\) \(1.34235 + 0.933680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (3.43 - 1.08i)T \)
good2 \( 1 + (-1.47 - 1.47i)T + 2iT^{2} \)
5 \( 1 + (-0.955 - 0.955i)T + 5iT^{2} \)
7 \( 1 + (3.08 + 3.08i)T + 7iT^{2} \)
11 \( 1 + (0.955 - 0.955i)T - 11iT^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + (-2.43 + 2.43i)T - 19iT^{2} \)
23 \( 1 + 7.81T + 23T^{2} \)
29 \( 1 - 8.23iT - 29T^{2} \)
31 \( 1 + (-1.73 + 1.73i)T - 31iT^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + (3.28 + 3.28i)T + 41iT^{2} \)
43 \( 1 - 7.46iT - 43T^{2} \)
47 \( 1 + (0.955 - 0.955i)T - 47iT^{2} \)
53 \( 1 - 5.48iT - 53T^{2} \)
59 \( 1 + (-1.57 + 1.57i)T - 59iT^{2} \)
61 \( 1 + 6.87T + 61T^{2} \)
67 \( 1 + (-2.91 + 2.91i)T - 67iT^{2} \)
71 \( 1 + (-6.23 - 6.23i)T + 71iT^{2} \)
73 \( 1 + (3.82 + 3.82i)T + 73iT^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 + (10.0 + 10.0i)T + 83iT^{2} \)
89 \( 1 + (-6.85 + 6.85i)T - 89iT^{2} \)
97 \( 1 + (-6.52 + 6.52i)T - 97iT^{2} \)
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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

−14.02024826549488559374229078602, −13.03918828443147879837926495822, −12.18322401709705527450922894613, −10.32154345082812789266331865808, −9.775653167225642428141429627942, −7.66848357119290992019915149164, −6.93169758139210494808962911317, −5.98902572207786961220625038547, −4.60097202377120032339031838679, −3.26394846747819685571058263561, 2.33030289990812373547848161442, 3.52098224145878854715955476349, 5.26890700708860102735047370830, 5.96039417451651055657905869382, 8.002129475080420375544429086315, 9.610405490322028647077881596891, 10.14103304740844009325849360763, 11.86545484356855022463471658971, 12.23826561088170084057280844939, 13.12367667790067052567543600198

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