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Properties

Label 2-117-117.25-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.999 + 0.0229i$
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  + (−0.929 − 0.536i)2-s + (−0.744 − 1.56i)3-s + (−0.423 − 0.733i)4-s + (−1.10 + 0.638i)5-s + (−0.147 + 1.85i)6-s + (−0.890 − 0.514i)7-s + 3.05i·8-s + (−1.89 + 2.32i)9-s + 1.37·10-s + (−4.03 − 2.33i)11-s + (−0.831 + 1.20i)12-s + (2.29 − 2.77i)13-s + (0.552 + 0.956i)14-s + (1.82 + 1.25i)15-s + (0.794 − 1.37i)16-s − 0.476·17-s + ⋯
L(s)  = 1  + (−0.657 − 0.379i)2-s + (−0.429 − 0.902i)3-s + (−0.211 − 0.366i)4-s + (−0.494 + 0.285i)5-s + (−0.0602 + 0.756i)6-s + (−0.336 − 0.194i)7-s + 1.08i·8-s + (−0.630 + 0.776i)9-s + 0.433·10-s + (−1.21 − 0.702i)11-s + (−0.240 + 0.348i)12-s + (0.637 − 0.770i)13-s + (0.147 + 0.255i)14-s + (0.470 + 0.323i)15-s + (0.198 − 0.344i)16-s − 0.115·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.999 + 0.0229i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.999 + 0.0229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00406366 - 0.354491i\)
\(L(\frac12)\) \(\approx\) \(0.00406366 - 0.354491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.744 + 1.56i)T \)
13 \( 1 + (-2.29 + 2.77i)T \)
good2 \( 1 + (0.929 + 0.536i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.10 - 0.638i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.890 + 0.514i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.03 + 2.33i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.476T + 17T^{2} \)
19 \( 1 + 6.69iT - 19T^{2} \)
23 \( 1 + (0.479 + 0.831i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.68 + 8.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.66 + 0.963i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.94iT - 37T^{2} \)
41 \( 1 + (-1.31 + 0.762i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.31 - 2.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.92 + 3.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.582T + 53T^{2} \)
59 \( 1 + (-3.64 + 2.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.71 - 8.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.01 + 1.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + (-6.45 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.86 - 5.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.85iT - 89T^{2} \)
97 \( 1 + (-14.9 - 8.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.27146479839547112420057908282, −11.63677796525137708686243627845, −10.99281481827203080697054975390, −10.10244844319366097153866197163, −8.531935831581019726093918670248, −7.78628089420482503990761837681, −6.33266655504946320374128419827, −5.14359834264630205777446493130, −2.73535719804955112267915682791, −0.49801548062322619528807714181, 3.57573180191110194802208780593, 4.76373210523942172655253227736, 6.33966400025001686200110119595, 7.79696557528602302388448689288, 8.739595920772435876039900361536, 9.759210770572216062166277644827, 10.62960510651221389809108492552, 12.05952600772119765578667293123, 12.72469528412278761826623006668, 14.21147626893074935690491266604

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