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Properties

Label 2-117-9.2-c2-0-14
Degree 22
Conductor 117117
Sign 0.996−0.0848i0.996 - 0.0848i
Analytic cond. 3.188013.18801
Root an. cond. 1.785501.78550
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.658 + 0.379i)2-s + (1.82 + 2.38i)3-s + (−1.71 − 2.96i)4-s + (7.70 − 4.45i)5-s + (0.294 + 2.26i)6-s + (−0.0636 + 0.110i)7-s − 5.64i·8-s + (−2.34 + 8.68i)9-s + 6.76·10-s + (−4.42 − 2.55i)11-s + (3.94 − 9.48i)12-s + (1.80 + 3.12i)13-s + (−0.0837 + 0.0483i)14-s + (24.6 + 10.2i)15-s + (−4.70 + 8.14i)16-s + 23.2i·17-s + ⋯
L(s)  = 1  + (0.329 + 0.189i)2-s + (0.607 + 0.794i)3-s + (−0.427 − 0.740i)4-s + (1.54 − 0.890i)5-s + (0.0491 + 0.376i)6-s + (−0.00909 + 0.0157i)7-s − 0.705i·8-s + (−0.261 + 0.965i)9-s + 0.676·10-s + (−0.402 − 0.232i)11-s + (0.328 − 0.790i)12-s + (0.138 + 0.240i)13-s + (−0.00598 + 0.00345i)14-s + (1.64 + 0.683i)15-s + (−0.293 + 0.508i)16-s + 1.36i·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.996−0.0848i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0848i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.996−0.0848i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0848i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.996−0.0848i0.996 - 0.0848i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(92,⋅)\chi_{117} (92, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.996−0.0848i)(2,\ 117,\ (\ :1),\ 0.996 - 0.0848i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.07783+0.0882852i2.07783 + 0.0882852i
L(12)L(\frac12) ≈\approx 2.07783+0.0882852i2.07783 + 0.0882852i
L(2)L(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+(−1.82−2.38i)T 1 + (-1.82 - 2.38i)T
13 1+(−1.80−3.12i)T 1 + (-1.80 - 3.12i)T
good2 1+(−0.658−0.379i)T+(2+3.46i)T2 1 + (-0.658 - 0.379i)T + (2 + 3.46i)T^{2}
5 1+(−7.70+4.45i)T+(12.5−21.6i)T2 1 + (-7.70 + 4.45i)T + (12.5 - 21.6i)T^{2}
7 1+(0.0636−0.110i)T+(−24.5−42.4i)T2 1 + (0.0636 - 0.110i)T + (-24.5 - 42.4i)T^{2}
11 1+(4.42+2.55i)T+(60.5+104.i)T2 1 + (4.42 + 2.55i)T + (60.5 + 104. i)T^{2}
17 1−23.2iT−289T2 1 - 23.2iT - 289T^{2}
19 1−9.50T+361T2 1 - 9.50T + 361T^{2}
23 1+(22.0−12.7i)T+(264.5−458.i)T2 1 + (22.0 - 12.7i)T + (264.5 - 458. i)T^{2}
29 1+(20.0+11.5i)T+(420.5+728.i)T2 1 + (20.0 + 11.5i)T + (420.5 + 728. i)T^{2}
31 1+(15.4+26.8i)T+(−480.5+832.i)T2 1 + (15.4 + 26.8i)T + (-480.5 + 832. i)T^{2}
37 1−7.35T+1.36e3T2 1 - 7.35T + 1.36e3T^{2}
41 1+(−32.4+18.7i)T+(840.5−1.45e3i)T2 1 + (-32.4 + 18.7i)T + (840.5 - 1.45e3i)T^{2}
43 1+(8.05−13.9i)T+(−924.5−1.60e3i)T2 1 + (8.05 - 13.9i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(66.9+38.6i)T+(1.10e3+1.91e3i)T2 1 + (66.9 + 38.6i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−49.0iT−2.80e3T2 1 - 49.0iT - 2.80e3T^{2}
59 1+(40.3−23.3i)T+(1.74e3−3.01e3i)T2 1 + (40.3 - 23.3i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−57.3+99.3i)T+(−1.86e3−3.22e3i)T2 1 + (-57.3 + 99.3i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−66.4−115.i)T+(−2.24e3+3.88e3i)T2 1 + (-66.4 - 115. i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+72.0iT−5.04e3T2 1 + 72.0iT - 5.04e3T^{2}
73 1−18.8T+5.32e3T2 1 - 18.8T + 5.32e3T^{2}
79 1+(0.111−0.193i)T+(−3.12e3−5.40e3i)T2 1 + (0.111 - 0.193i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−50.8−29.3i)T+(3.44e3+5.96e3i)T2 1 + (-50.8 - 29.3i)T + (3.44e3 + 5.96e3i)T^{2}
89 1−100.iT−7.92e3T2 1 - 100. iT - 7.92e3T^{2}
97 1+(−4.44+7.69i)T+(−4.70e3−8.14e3i)T2 1 + (-4.44 + 7.69i)T + (-4.70e3 - 8.14e3i)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

−13.53133237637794225438307547172, −12.82795874499634628804751470104, −10.82688769066998586372393699730, −9.809004070722368301296865940357, −9.372297521455157161238248073259, −8.241865770152329023661088847810, −5.99754096297738404441083600434, −5.36357211210748377459527720640, −4.09552885431623171307839907861, −1.85799524516376930070861282219, 2.23854030147246654969040221107, 3.22233761427317094423817511078, 5.34070827353504039604506403099, 6.70679211777142930532791318104, 7.72066637654356085899349741018, 9.057898875487896390199161685354, 9.942828283384208090393479181858, 11.39479559524943314738933105058, 12.62736198962073009417626886881, 13.37598291331939246532023474547

Graph of the ZZ-function along the critical line