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Properties

Label 2-117-117.59-c1-0-4
Degree 22
Conductor 117117
Sign −0.386−0.922i-0.386 - 0.922i
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  + (−0.754 + 0.754i)2-s + (0.861 + 1.50i)3-s + 0.862i·4-s + (0.595 + 2.22i)5-s + (−1.78 − 0.483i)6-s + (−1.34 − 5.00i)7-s + (−2.15 − 2.15i)8-s + (−1.51 + 2.58i)9-s + (−2.12 − 1.22i)10-s + (1.93 + 1.93i)11-s + (−1.29 + 0.743i)12-s + (3.42 + 1.13i)13-s + (4.78 + 2.76i)14-s + (−2.82 + 2.80i)15-s + 1.52·16-s + (0.0716 + 0.124i)17-s + ⋯
L(s)  = 1  + (−0.533 + 0.533i)2-s + (0.497 + 0.867i)3-s + 0.431i·4-s + (0.266 + 0.993i)5-s + (−0.727 − 0.197i)6-s + (−0.506 − 1.89i)7-s + (−0.763 − 0.763i)8-s + (−0.505 + 0.862i)9-s + (−0.671 − 0.387i)10-s + (0.583 + 0.583i)11-s + (−0.374 + 0.214i)12-s + (0.949 + 0.315i)13-s + (1.27 + 0.737i)14-s + (−0.729 + 0.725i)15-s + 0.382·16-s + (0.0173 + 0.0300i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.386−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((−0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.386−0.922i-0.386 - 0.922i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(59,⋅)\chi_{117} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), −0.386−0.922i)(2,\ 117,\ (\ :1/2),\ -0.386 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 0.512875+0.771147i0.512875 + 0.771147i
L(12)L(\frac12) ≈\approx 0.512875+0.771147i0.512875 + 0.771147i
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+(−0.861−1.50i)T 1 + (-0.861 - 1.50i)T
13 1+(−3.42−1.13i)T 1 + (-3.42 - 1.13i)T
good2 1+(0.754−0.754i)T−2iT2 1 + (0.754 - 0.754i)T - 2iT^{2}
5 1+(−0.595−2.22i)T+(−4.33+2.5i)T2 1 + (-0.595 - 2.22i)T + (-4.33 + 2.5i)T^{2}
7 1+(1.34+5.00i)T+(−6.06+3.5i)T2 1 + (1.34 + 5.00i)T + (-6.06 + 3.5i)T^{2}
11 1+(−1.93−1.93i)T+11iT2 1 + (-1.93 - 1.93i)T + 11iT^{2}
17 1+(−0.0716−0.124i)T+(−8.5+14.7i)T2 1 + (-0.0716 - 0.124i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.552+2.06i)T+(−16.4−9.5i)T2 1 + (-0.552 + 2.06i)T + (-16.4 - 9.5i)T^{2}
23 1+(−0.421−0.730i)T+(−11.5+19.9i)T2 1 + (-0.421 - 0.730i)T + (-11.5 + 19.9i)T^{2}
29 1+0.605iT−29T2 1 + 0.605iT - 29T^{2}
31 1+(−3.38+0.905i)T+(26.8−15.5i)T2 1 + (-3.38 + 0.905i)T + (26.8 - 15.5i)T^{2}
37 1+(−0.973−3.63i)T+(−32.0+18.5i)T2 1 + (-0.973 - 3.63i)T + (-32.0 + 18.5i)T^{2}
41 1+(7.55+2.02i)T+(35.5+20.5i)T2 1 + (7.55 + 2.02i)T + (35.5 + 20.5i)T^{2}
43 1+(−0.187−0.108i)T+(21.5+37.2i)T2 1 + (-0.187 - 0.108i)T + (21.5 + 37.2i)T^{2}
47 1+(−2.72+10.1i)T+(−40.7−23.5i)T2 1 + (-2.72 + 10.1i)T + (-40.7 - 23.5i)T^{2}
53 1+8.00iT−53T2 1 + 8.00iT - 53T^{2}
59 1+(−5.26−5.26i)T+59iT2 1 + (-5.26 - 5.26i)T + 59iT^{2}
61 1+(−0.675+1.16i)T+(−30.5−52.8i)T2 1 + (-0.675 + 1.16i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.98+7.40i)T+(−58.0−33.5i)T2 1 + (-1.98 + 7.40i)T + (-58.0 - 33.5i)T^{2}
71 1+(7.04+1.88i)T+(61.4+35.5i)T2 1 + (7.04 + 1.88i)T + (61.4 + 35.5i)T^{2}
73 1+(3.76−3.76i)T−73iT2 1 + (3.76 - 3.76i)T - 73iT^{2}
79 1+(−1.62−2.81i)T+(−39.5+68.4i)T2 1 + (-1.62 - 2.81i)T + (-39.5 + 68.4i)T^{2}
83 1+(14.7+3.94i)T+(71.8+41.5i)T2 1 + (14.7 + 3.94i)T + (71.8 + 41.5i)T^{2}
89 1+(14.2−3.81i)T+(77.0−44.5i)T2 1 + (14.2 - 3.81i)T + (77.0 - 44.5i)T^{2}
97 1+(−0.525+0.140i)T+(84.0−48.5i)T2 1 + (-0.525 + 0.140i)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

−13.91409582687206723862681798471, −13.28288100563700021712618338641, −11.43812964724415583611798484663, −10.34963493588358526722925164732, −9.743807966769283244867501096207, −8.495493695927505328107910845822, −7.22241714939206423589860825790, −6.59735196518691770750212428303, −4.16052885113244246093033501751, −3.29681143037659109953533805763, 1.40549995222116551758196513686, 2.88934398357885621253360214681, 5.59030180734161502606015864778, 6.21617063047995649081568043890, 8.521711729460237142311823728961, 8.774456793535409894435392715638, 9.659699739088766894812532768496, 11.38144360865399729646568312646, 12.20654539758846547435563157050, 12.94874294446718611082489415511

Graph of the ZZ-function along the critical line