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Properties

Label 2-117-9.7-c1-0-6
Degree 22
Conductor 117117
Sign 0.991+0.133i0.991 + 0.133i
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  + (−1.39 + 2.42i)2-s + (0.413 − 1.68i)3-s + (−2.90 − 5.03i)4-s + (−0.413 − 0.717i)5-s + (3.49 + 3.35i)6-s + (1.24 − 2.15i)7-s + 10.6·8-s + (−2.65 − 1.39i)9-s + 2.31·10-s + (0.459 − 0.796i)11-s + (−9.67 + 2.80i)12-s + (−0.5 − 0.866i)13-s + (3.47 + 6.01i)14-s + (−1.37 + 0.399i)15-s + (−9.08 + 15.7i)16-s − 1.53·17-s + ⋯
L(s)  = 1  + (−0.988 + 1.71i)2-s + (0.239 − 0.971i)3-s + (−1.45 − 2.51i)4-s + (−0.185 − 0.320i)5-s + (1.42 + 1.36i)6-s + (0.469 − 0.813i)7-s + 3.76·8-s + (−0.885 − 0.464i)9-s + 0.731·10-s + (0.138 − 0.240i)11-s + (−2.79 + 0.809i)12-s + (−0.138 − 0.240i)13-s + (0.928 + 1.60i)14-s + (−0.355 + 0.103i)15-s + (−2.27 + 3.93i)16-s − 0.372·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.991+0.133i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.991+0.133i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.991+0.133i0.991 + 0.133i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(79,⋅)\chi_{117} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.991+0.133i)(2,\ 117,\ (\ :1/2),\ 0.991 + 0.133i)

Particular Values

L(1)L(1) ≈\approx 0.618227−0.0413731i0.618227 - 0.0413731i
L(12)L(\frac12) ≈\approx 0.618227−0.0413731i0.618227 - 0.0413731i
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+(−0.413+1.68i)T 1 + (-0.413 + 1.68i)T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(1.39−2.42i)T+(−1−1.73i)T2 1 + (1.39 - 2.42i)T + (-1 - 1.73i)T^{2}
5 1+(0.413+0.717i)T+(−2.5+4.33i)T2 1 + (0.413 + 0.717i)T + (-2.5 + 4.33i)T^{2}
7 1+(−1.24+2.15i)T+(−3.5−6.06i)T2 1 + (-1.24 + 2.15i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.459+0.796i)T+(−5.5−9.52i)T2 1 + (-0.459 + 0.796i)T + (-5.5 - 9.52i)T^{2}
17 1+1.53T+17T2 1 + 1.53T + 17T^{2}
19 1−3.40T+19T2 1 - 3.40T + 19T^{2}
23 1+(0.490+0.850i)T+(−11.5+19.9i)T2 1 + (0.490 + 0.850i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.595+1.03i)T+(−14.5−25.1i)T2 1 + (-0.595 + 1.03i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.12−5.41i)T+(−15.5+26.8i)T2 1 + (-3.12 - 5.41i)T + (-15.5 + 26.8i)T^{2}
37 1−1.83T+37T2 1 - 1.83T + 37T^{2}
41 1+(−2.86−4.96i)T+(−20.5+35.5i)T2 1 + (-2.86 - 4.96i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.792−1.37i)T+(−21.5−37.2i)T2 1 + (0.792 - 1.37i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.68−4.65i)T+(−23.5−40.7i)T2 1 + (2.68 - 4.65i)T + (-23.5 - 40.7i)T^{2}
53 1+5.82T+53T2 1 + 5.82T + 53T^{2}
59 1+(−0.477−0.826i)T+(−29.5+51.0i)T2 1 + (-0.477 - 0.826i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.44+7.70i)T+(−30.5−52.8i)T2 1 + (-4.44 + 7.70i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.56−11.3i)T+(−33.5+58.0i)T2 1 + (-6.56 - 11.3i)T + (-33.5 + 58.0i)T^{2}
71 1−5.27T+71T2 1 - 5.27T + 71T^{2}
73 1+10.0T+73T2 1 + 10.0T + 73T^{2}
79 1+(5.06−8.76i)T+(−39.5−68.4i)T2 1 + (5.06 - 8.76i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.16+2.01i)T+(−41.5−71.8i)T2 1 + (-1.16 + 2.01i)T + (-41.5 - 71.8i)T^{2}
89 1−12.5T+89T2 1 - 12.5T + 89T^{2}
97 1+(−3.94+6.82i)T+(−48.5−84.0i)T2 1 + (-3.94 + 6.82i)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

−14.03712003881681358605990507094, −12.93643390873539059849961019663, −11.19949747110582826153274092743, −9.915154330202915091074058544069, −8.684877030031057036131710095507, −7.972609760099862616376290886388, −7.13162342158559728392625896078, −6.15119559422917816639441724312, −4.74503865279197543189741794207, −1.01232277694787782936942913152, 2.31882715963563022198515939422, 3.58507550871379443115494966123, 4.89681319362438311179293979568, 7.68389118924110958399288259982, 8.777382190259528145621716692589, 9.456040581122151296448786285857, 10.42048797130641020483316563391, 11.39699797267927088397137993667, 11.89757374288909569681091093284, 13.24954386025497884613330836545

Graph of the ZZ-function along the critical line