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Properties

Label 2-117-39.32-c1-0-0
Degree $2$
Conductor $117$
Sign $0.0660 - 0.997i$
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  + (−1.93 + 0.517i)2-s + (1.73 − 0.999i)4-s + (0.517 + 0.517i)5-s + (−0.5 + 1.86i)7-s + (−1.26 − 0.732i)10-s + (1.60 + 5.98i)11-s + (−1.59 + 3.23i)13-s − 3.86i·14-s + (−1.99 + 3.46i)16-s + (−0.896 − 1.55i)17-s + (3.36 + 0.901i)19-s + (1.41 + 0.378i)20-s + (−6.19 − 10.7i)22-s + (1.22 − 2.12i)23-s − 4.46i·25-s + (1.41 − 7.07i)26-s + ⋯
L(s)  = 1  + (−1.36 + 0.366i)2-s + (0.866 − 0.499i)4-s + (0.231 + 0.231i)5-s + (−0.188 + 0.705i)7-s + (−0.400 − 0.231i)10-s + (0.483 + 1.80i)11-s + (−0.443 + 0.896i)13-s − 1.03i·14-s + (−0.499 + 0.866i)16-s + (−0.217 − 0.376i)17-s + (0.772 + 0.206i)19-s + (0.316 + 0.0847i)20-s + (−1.32 − 2.28i)22-s + (0.255 − 0.442i)23-s − 0.892i·25-s + (0.277 − 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0660 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0660 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0660 - 0.997i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.0660 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391198 + 0.366161i\)
\(L(\frac12)\) \(\approx\) \(0.391198 + 0.366161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 3.23i)T \)
good2 \( 1 + (1.93 - 0.517i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.517 - 0.517i)T + 5iT^{2} \)
7 \( 1 + (0.5 - 1.86i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.60 - 5.98i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.896 + 1.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.36 - 0.901i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 0.707i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.36 + 6.36i)T - 31iT^{2} \)
37 \( 1 + (7.83 - 2.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.27 - 1.41i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.69 + 2.13i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.76 - 4.76i)T - 47iT^{2} \)
53 \( 1 + 8.76iT - 53T^{2} \)
59 \( 1 + (-9.84 - 2.63i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.40 + 2.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.40 + 8.96i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.429 + 1.60i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.09 - 8.09i)T + 73iT^{2} \)
79 \( 1 + 3.19T + 79T^{2} \)
83 \( 1 + (0.378 + 0.378i)T + 83iT^{2} \)
89 \( 1 + (2.96 + 11.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.6 + 3.13i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.97321526856322398312583050941, −12.50337464855885317219944053850, −11.62513103204136021849264473028, −10.02641373009283057941136604072, −9.627975629817276525987451343493, −8.578556217750688017452875500983, −7.26832852103353012770451142622, −6.51608986447704089239031146315, −4.60178722560852836747103706837, −2.09688643993328007616577316613, 0.985285673803616486930511727499, 3.26183312459923570166888887902, 5.38849914120836813171592587078, 7.00541497590791188497677988418, 8.218843342029775255855556441437, 9.013661656773527708194221130191, 10.13401539150428374327224732312, 10.87580071620580864653886701836, 11.86583149699983186822268019918, 13.39091117282503235143442785145

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