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Properties

Label 2-117-117.25-c1-0-8
Degree $2$
Conductor $117$
Sign $0.856 - 0.515i$
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  + (1.97 + 1.14i)2-s + (0.833 − 1.51i)3-s + (1.60 + 2.78i)4-s + (−2.78 + 1.60i)5-s + (3.38 − 2.05i)6-s + (−2.09 − 1.21i)7-s + 2.76i·8-s + (−1.61 − 2.53i)9-s − 7.34·10-s + (1.27 + 0.737i)11-s + (5.56 − 0.120i)12-s + (3.56 − 0.535i)13-s + (−2.76 − 4.79i)14-s + (0.121 + 5.56i)15-s + (0.0535 − 0.0927i)16-s − 5.12·17-s + ⋯
L(s)  = 1  + (1.39 + 0.807i)2-s + (0.481 − 0.876i)3-s + (0.803 + 1.39i)4-s + (−1.24 + 0.719i)5-s + (1.38 − 0.837i)6-s + (−0.793 − 0.457i)7-s + 0.978i·8-s + (−0.537 − 0.843i)9-s − 2.32·10-s + (0.385 + 0.222i)11-s + (1.60 − 0.0348i)12-s + (0.988 − 0.148i)13-s + (−0.739 − 1.28i)14-s + (0.0312 + 1.43i)15-s + (0.0133 − 0.0231i)16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.856 - 0.515i$
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: \((2,\ 117,\ (\ :1/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82236 + 0.505916i\)
\(L(\frac12)\) \(\approx\) \(1.82236 + 0.505916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.833 + 1.51i)T \)
13 \( 1 + (-3.56 + 0.535i)T \)
good2 \( 1 + (-1.97 - 1.14i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (2.78 - 1.60i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.09 + 1.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.27 - 0.737i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 - 1.13iT - 19T^{2} \)
23 \( 1 + (-4.61 - 7.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.487 - 0.844i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.16 - 1.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.22iT - 37T^{2} \)
41 \( 1 + (-3.47 + 2.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.33 - 0.769i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.739T + 53T^{2} \)
59 \( 1 + (6.72 - 3.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.669 - 0.386i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.01iT - 71T^{2} \)
73 \( 1 + 9.21iT - 73T^{2} \)
79 \( 1 + (1.86 - 3.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.3 + 7.13i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.21iT - 89T^{2} \)
97 \( 1 + (-13.1 - 7.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.58430053051277064825048880592, −12.99682948074297444544998103129, −11.96894565183473380147976392870, −11.01626144175725339017100190876, −8.958163451702020805830315732777, −7.43749802172832334842752583116, −7.06646663725221935626487832689, −6.00271817754724616364791287270, −3.95800541903625752442825210372, −3.27378127595196457363195219638, 2.89458481365581741435447923099, 4.02159286623981376501129711249, 4.72118637319553475088406179470, 6.27970062049964989363454033651, 8.381567351588612031252404677126, 9.200759368453928788001361973297, 10.87064342100046082654505515998, 11.40385447722023910753167433372, 12.59185073742298700609078090077, 13.22247352657698431461065746495

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