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Properties

Label 2-117-117.32-c1-0-3
Degree 22
Conductor 117117
Sign 0.678+0.734i0.678 + 0.734i
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  + (−1.53 − 1.53i)2-s + (1.40 + 1.00i)3-s + 2.71i·4-s + (−0.529 + 0.142i)5-s + (−0.619 − 3.71i)6-s + (4.13 − 1.10i)7-s + (1.10 − 1.10i)8-s + (0.974 + 2.83i)9-s + (1.03 + 0.595i)10-s + (2.64 − 2.64i)11-s + (−2.73 + 3.83i)12-s + (−3.49 − 0.872i)13-s + (−8.06 − 4.65i)14-s + (−0.889 − 0.333i)15-s + 2.04·16-s + (−0.784 − 1.35i)17-s + ⋯
L(s)  = 1  + (−1.08 − 1.08i)2-s + (0.813 + 0.581i)3-s + 1.35i·4-s + (−0.237 + 0.0635i)5-s + (−0.252 − 1.51i)6-s + (1.56 − 0.419i)7-s + (0.389 − 0.389i)8-s + (0.324 + 0.945i)9-s + (0.326 + 0.188i)10-s + (0.798 − 0.798i)11-s + (−0.789 + 1.10i)12-s + (−0.970 − 0.242i)13-s + (−2.15 − 1.24i)14-s + (−0.229 − 0.0860i)15-s + 0.512·16-s + (−0.190 − 0.329i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.678+0.734i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.678+0.734i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.678+0.734i0.678 + 0.734i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(32,⋅)\chi_{117} (32, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.678+0.734i)(2,\ 117,\ (\ :1/2),\ 0.678 + 0.734i)

Particular Values

L(1)L(1) ≈\approx 0.778215−0.340557i0.778215 - 0.340557i
L(12)L(\frac12) ≈\approx 0.778215−0.340557i0.778215 - 0.340557i
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+(−1.40−1.00i)T 1 + (-1.40 - 1.00i)T
13 1+(3.49+0.872i)T 1 + (3.49 + 0.872i)T
good2 1+(1.53+1.53i)T+2iT2 1 + (1.53 + 1.53i)T + 2iT^{2}
5 1+(0.529−0.142i)T+(4.33−2.5i)T2 1 + (0.529 - 0.142i)T + (4.33 - 2.5i)T^{2}
7 1+(−4.13+1.10i)T+(6.06−3.5i)T2 1 + (-4.13 + 1.10i)T + (6.06 - 3.5i)T^{2}
11 1+(−2.64+2.64i)T−11iT2 1 + (-2.64 + 2.64i)T - 11iT^{2}
17 1+(0.784+1.35i)T+(−8.5+14.7i)T2 1 + (0.784 + 1.35i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.27+0.876i)T+(16.4+9.5i)T2 1 + (3.27 + 0.876i)T + (16.4 + 9.5i)T^{2}
23 1+(−2.51−4.35i)T+(−11.5+19.9i)T2 1 + (-2.51 - 4.35i)T + (-11.5 + 19.9i)T^{2}
29 1−1.18iT−29T2 1 - 1.18iT - 29T^{2}
31 1+(−0.395−1.47i)T+(−26.8+15.5i)T2 1 + (-0.395 - 1.47i)T + (-26.8 + 15.5i)T^{2}
37 1+(−1.31+0.351i)T+(32.0−18.5i)T2 1 + (-1.31 + 0.351i)T + (32.0 - 18.5i)T^{2}
41 1+(1.37−5.13i)T+(−35.5−20.5i)T2 1 + (1.37 - 5.13i)T + (-35.5 - 20.5i)T^{2}
43 1+(8.94+5.16i)T+(21.5+37.2i)T2 1 + (8.94 + 5.16i)T + (21.5 + 37.2i)T^{2}
47 1+(6.81+1.82i)T+(40.7+23.5i)T2 1 + (6.81 + 1.82i)T + (40.7 + 23.5i)T^{2}
53 1+5.04iT−53T2 1 + 5.04iT - 53T^{2}
59 1+(7.65−7.65i)T−59iT2 1 + (7.65 - 7.65i)T - 59iT^{2}
61 1+(−2.96+5.13i)T+(−30.5−52.8i)T2 1 + (-2.96 + 5.13i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.07+0.824i)T+(58.0+33.5i)T2 1 + (3.07 + 0.824i)T + (58.0 + 33.5i)T^{2}
71 1+(−0.774+2.89i)T+(−61.4−35.5i)T2 1 + (-0.774 + 2.89i)T + (-61.4 - 35.5i)T^{2}
73 1+(9.10+9.10i)T+73iT2 1 + (9.10 + 9.10i)T + 73iT^{2}
79 1+(−7.18−12.4i)T+(−39.5+68.4i)T2 1 + (-7.18 - 12.4i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.67+13.7i)T+(−71.8−41.5i)T2 1 + (-3.67 + 13.7i)T + (-71.8 - 41.5i)T^{2}
89 1+(−1.63−6.09i)T+(−77.0+44.5i)T2 1 + (-1.63 - 6.09i)T + (-77.0 + 44.5i)T^{2}
97 1+(−0.846−3.15i)T+(−84.0+48.5i)T2 1 + (-0.846 - 3.15i)T + (-84.0 + 48.5i)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

−13.40944706664091697988380616754, −11.75230014235551762159817397363, −11.18781987771255746124929762071, −10.25498700735143321132836274844, −9.197426063021747055906968166717, −8.353327163484507768697842653989, −7.53248875165629719819827787184, −4.89035273629691310133586006798, −3.42408714322963958336424617801, −1.79837603557507823888219416011, 1.87150160434170496134822466049, 4.54340107145309828383569111305, 6.39490757455415737710794210149, 7.44041985590458411990296141876, 8.212488847181904696611719568399, 8.934848277101174049164032828983, 10.01558120162293991685267058810, 11.67986184832326708543839009034, 12.59282068695968014774818317414, 14.30381050014522807643305574415

Graph of the ZZ-function along the critical line