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Properties

Label 2-117-117.83-c1-0-5
Degree 22
Conductor 117117
Sign 0.762+0.646i0.762 + 0.646i
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  + (−0.999 + 0.267i)2-s + (−1.73 + 0.0358i)3-s + (−0.805 + 0.465i)4-s + (0.483 − 1.80i)5-s + (1.72 − 0.499i)6-s + (3.99 − 1.07i)7-s + (2.14 − 2.14i)8-s + (2.99 − 0.124i)9-s + 1.93i·10-s + (−0.995 − 3.71i)11-s + (1.37 − 0.834i)12-s + (−2.74 + 2.34i)13-s + (−3.70 + 2.13i)14-s + (−0.772 + 3.14i)15-s + (−0.636 + 1.10i)16-s − 0.582·17-s + ⋯
L(s)  = 1  + (−0.706 + 0.189i)2-s + (−0.999 + 0.0206i)3-s + (−0.402 + 0.232i)4-s + (0.216 − 0.806i)5-s + (0.702 − 0.203i)6-s + (1.51 − 0.404i)7-s + (0.757 − 0.757i)8-s + (0.999 − 0.0413i)9-s + 0.610i·10-s + (−0.300 − 1.12i)11-s + (0.397 − 0.240i)12-s + (−0.760 + 0.649i)13-s + (−0.990 + 0.571i)14-s + (−0.199 + 0.810i)15-s + (−0.159 + 0.275i)16-s − 0.141·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.762+0.646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.762+0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.762+0.646i0.762 + 0.646i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(83,⋅)\chi_{117} (83, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.762+0.646i)(2,\ 117,\ (\ :1/2),\ 0.762 + 0.646i)

Particular Values

L(1)L(1) ≈\approx 0.528918−0.194040i0.528918 - 0.194040i
L(12)L(\frac12) ≈\approx 0.528918−0.194040i0.528918 - 0.194040i
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+(1.73−0.0358i)T 1 + (1.73 - 0.0358i)T
13 1+(2.74−2.34i)T 1 + (2.74 - 2.34i)T
good2 1+(0.999−0.267i)T+(1.73−i)T2 1 + (0.999 - 0.267i)T + (1.73 - i)T^{2}
5 1+(−0.483+1.80i)T+(−4.33−2.5i)T2 1 + (-0.483 + 1.80i)T + (-4.33 - 2.5i)T^{2}
7 1+(−3.99+1.07i)T+(6.06−3.5i)T2 1 + (-3.99 + 1.07i)T + (6.06 - 3.5i)T^{2}
11 1+(0.995+3.71i)T+(−9.52+5.5i)T2 1 + (0.995 + 3.71i)T + (-9.52 + 5.5i)T^{2}
17 1+0.582T+17T2 1 + 0.582T + 17T^{2}
19 1+(−4.41+4.41i)T−19iT2 1 + (-4.41 + 4.41i)T - 19iT^{2}
23 1+(2.65+4.59i)T+(−11.5+19.9i)T2 1 + (2.65 + 4.59i)T + (-11.5 + 19.9i)T^{2}
29 1+(−5.19−2.99i)T+(14.5+25.1i)T2 1 + (-5.19 - 2.99i)T + (14.5 + 25.1i)T^{2}
31 1+(−2.14−0.574i)T+(26.8+15.5i)T2 1 + (-2.14 - 0.574i)T + (26.8 + 15.5i)T^{2}
37 1+(4.32+4.32i)T+37iT2 1 + (4.32 + 4.32i)T + 37iT^{2}
41 1+(1.64−6.13i)T+(−35.5−20.5i)T2 1 + (1.64 - 6.13i)T + (-35.5 - 20.5i)T^{2}
43 1+(5.20+3.00i)T+(21.5+37.2i)T2 1 + (5.20 + 3.00i)T + (21.5 + 37.2i)T^{2}
47 1+(−1.64−6.15i)T+(−40.7+23.5i)T2 1 + (-1.64 - 6.15i)T + (-40.7 + 23.5i)T^{2}
53 1−1.69iT−53T2 1 - 1.69iT - 53T^{2}
59 1+(−6.45−1.72i)T+(51.0+29.5i)T2 1 + (-6.45 - 1.72i)T + (51.0 + 29.5i)T^{2}
61 1+(1.12−1.95i)T+(−30.5−52.8i)T2 1 + (1.12 - 1.95i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.21+0.594i)T+(58.0+33.5i)T2 1 + (2.21 + 0.594i)T + (58.0 + 33.5i)T^{2}
71 1+(1.87+1.87i)T+71iT2 1 + (1.87 + 1.87i)T + 71iT^{2}
73 1+(−5.42−5.42i)T+73iT2 1 + (-5.42 - 5.42i)T + 73iT^{2}
79 1+(1.10−1.91i)T+(−39.5−68.4i)T2 1 + (1.10 - 1.91i)T + (-39.5 - 68.4i)T^{2}
83 1+(11.0−2.95i)T+(71.8−41.5i)T2 1 + (11.0 - 2.95i)T + (71.8 - 41.5i)T^{2}
89 1+(0.110−0.110i)T−89iT2 1 + (0.110 - 0.110i)T - 89iT^{2}
97 1+(−2.91−10.8i)T+(−84.0+48.5i)T2 1 + (-2.91 - 10.8i)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

−13.39623203088095927854221194753, −12.23207122724305890939816612742, −11.26702168548702435512682195439, −10.29186001307263671663492586746, −9.030877673593137118347038839978, −8.127953177035622772053393454698, −6.98946390628369568458231613005, −5.15611134518996228982837190762, −4.52309152563571893428071425357, −1.01644226386091684264134987962, 1.81866309116641564446070128451, 4.73399641485754030290635031191, 5.51765095770429577542983445021, 7.27397957618144287980638175779, 8.166349017335637991936781490414, 9.952280471567240716682528293655, 10.26460056443369381401910932966, 11.45905833322914168981876075054, 12.20641199345490123406904442383, 13.76820543998169707102905968192

Graph of the ZZ-function along the critical line