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Properties

Label 2-117-117.103-c1-0-8
Degree 22
Conductor 117117
Sign 0.856+0.515i0.856 + 0.515i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(117s/2 ΓC(s) L(s)=((0.856+0.515i) Λ‾(2−s)\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}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.856+0.515i) Λ‾(1−s)\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: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.856+0.515i0.856 + 0.515i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(103,⋅)\chi_{117} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.856+0.515i)(2,\ 117,\ (\ :1/2),\ 0.856 + 0.515i)

Particular Values

L(1)L(1) ≈\approx 1.82236−0.505916i1.82236 - 0.505916i
L(12)L(\frac12) ≈\approx 1.82236−0.505916i1.82236 - 0.505916i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.833−1.51i)T 1 + (-0.833 - 1.51i)T
13 1+(−3.56−0.535i)T 1 + (-3.56 - 0.535i)T
good2 1+(−1.97+1.14i)T+(1−1.73i)T2 1 + (-1.97 + 1.14i)T + (1 - 1.73i)T^{2}
5 1+(2.78+1.60i)T+(2.5+4.33i)T2 1 + (2.78 + 1.60i)T + (2.5 + 4.33i)T^{2}
7 1+(2.09−1.21i)T+(3.5−6.06i)T2 1 + (2.09 - 1.21i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.27+0.737i)T+(5.5−9.52i)T2 1 + (-1.27 + 0.737i)T + (5.5 - 9.52i)T^{2}
17 1+5.12T+17T2 1 + 5.12T + 17T^{2}
19 1+1.13iT−19T2 1 + 1.13iT - 19T^{2}
23 1+(−4.61+7.99i)T+(−11.5−19.9i)T2 1 + (-4.61 + 7.99i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.487+0.844i)T+(−14.5+25.1i)T2 1 + (0.487 + 0.844i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.16+1.82i)T+(15.5+26.8i)T2 1 + (3.16 + 1.82i)T + (15.5 + 26.8i)T^{2}
37 1−4.22iT−37T2 1 - 4.22iT - 37T^{2}
41 1+(−3.47−2.00i)T+(20.5+35.5i)T2 1 + (-3.47 - 2.00i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.33−7.50i)T+(−21.5+37.2i)T2 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.33+0.769i)T+(23.5−40.7i)T2 1 + (-1.33 + 0.769i)T + (23.5 - 40.7i)T^{2}
53 1−0.739T+53T2 1 - 0.739T + 53T^{2}
59 1+(6.72+3.88i)T+(29.5+51.0i)T2 1 + (6.72 + 3.88i)T + (29.5 + 51.0i)T^{2}
61 1+(4.06+7.04i)T+(−30.5+52.8i)T2 1 + (4.06 + 7.04i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.669+0.386i)T+(33.5+58.0i)T2 1 + (0.669 + 0.386i)T + (33.5 + 58.0i)T^{2}
71 1+3.01iT−71T2 1 + 3.01iT - 71T^{2}
73 1−9.21iT−73T2 1 - 9.21iT - 73T^{2}
79 1+(1.86+3.23i)T+(−39.5+68.4i)T2 1 + (1.86 + 3.23i)T + (-39.5 + 68.4i)T^{2}
83 1+(12.3−7.13i)T+(41.5−71.8i)T2 1 + (12.3 - 7.13i)T + (41.5 - 71.8i)T^{2}
89 1−8.21iT−89T2 1 - 8.21iT - 89T^{2}
97 1+(−13.1+7.61i)T+(48.5−84.0i)T2 1 + (-13.1 + 7.61i)T + (48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line