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Properties

Label 2-117-117.61-c1-0-9
Degree $2$
Conductor $117$
Sign $0.939 + 0.341i$
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  + (0.433 + 0.751i)2-s + (0.753 − 1.55i)3-s + (0.623 − 1.08i)4-s + (−0.0324 − 0.0561i)5-s + (1.49 − 0.110i)6-s − 3.92·7-s + 2.81·8-s + (−1.86 − 2.35i)9-s + (0.0281 − 0.0486i)10-s + (2.64 + 4.58i)11-s + (−1.21 − 1.78i)12-s + (−0.188 + 3.60i)13-s + (−1.70 − 2.94i)14-s + (−0.111 + 0.00822i)15-s + (−0.0259 − 0.0449i)16-s + (2.28 + 3.95i)17-s + ⋯
L(s)  = 1  + (0.306 + 0.531i)2-s + (0.435 − 0.900i)3-s + (0.311 − 0.540i)4-s + (−0.0144 − 0.0251i)5-s + (0.611 − 0.0449i)6-s − 1.48·7-s + 0.995·8-s + (−0.621 − 0.783i)9-s + (0.00888 − 0.0153i)10-s + (0.797 + 1.38i)11-s + (−0.350 − 0.515i)12-s + (−0.0523 + 0.998i)13-s + (−0.454 − 0.787i)14-s + (−0.0289 + 0.00212i)15-s + (−0.00649 − 0.0112i)16-s + (0.553 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.939 + 0.341i$
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: \((2,\ 117,\ (\ :1/2),\ 0.939 + 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33288 - 0.234554i\)
\(L(\frac12)\) \(\approx\) \(1.33288 - 0.234554i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.753 + 1.55i)T \)
13 \( 1 + (0.188 - 3.60i)T \)
good2 \( 1 + (-0.433 - 0.751i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0324 + 0.0561i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 + (-2.64 - 4.58i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.28 - 3.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.281 - 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 + (3.00 + 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.23 + 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.506 - 0.877i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.34T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + (2.22 - 3.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 + (4.57 - 7.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.35T + 61T^{2} \)
67 \( 1 - 3.53T + 67T^{2} \)
71 \( 1 + (5.02 + 8.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.39T + 73T^{2} \)
79 \( 1 + (-5.67 + 9.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.87 - 3.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.00 - 3.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.35T + 97T^{2} \)
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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

−13.53949360345732216435778570222, −12.63963697920475667842904001030, −11.76792856040891599805945181645, −10.04632161741693265605931762564, −9.315524661305340827030021727651, −7.63935797706896552429434298781, −6.65181668382427841203561598550, −6.10413471646410645338016961905, −4.03311299288503483690237201078, −1.98334675072236804226431561628, 3.20277590611810074356932693255, 3.45642556681621400368505097166, 5.41996117903678236095664800257, 6.99122519952595647424397324723, 8.441852933714119890601465291058, 9.486095326190331984094842495985, 10.53982532418988689512585395754, 11.43880804164290092191350761015, 12.62964993851995954690986549617, 13.48617464387701748908860034552

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