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Properties

Label 2-117-117.11-c1-0-10
Degree 22
Conductor 117117
Sign 0.0346+0.999i0.0346 + 0.999i
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  + (1.12 − 1.12i)2-s + (0.108 − 1.72i)3-s − 0.527i·4-s + (−1.83 − 0.492i)5-s + (−1.82 − 2.06i)6-s + (1.85 + 0.498i)7-s + (1.65 + 1.65i)8-s + (−2.97 − 0.376i)9-s + (−2.61 + 1.51i)10-s + (−1.22 − 1.22i)11-s + (−0.911 − 0.0573i)12-s + (2.63 + 2.46i)13-s + (2.64 − 1.52i)14-s + (−1.05 + 3.12i)15-s + 4.77·16-s + (−2.74 + 4.76i)17-s + ⋯
L(s)  = 1  + (0.794 − 0.794i)2-s + (0.0628 − 0.998i)3-s − 0.263i·4-s + (−0.821 − 0.220i)5-s + (−0.743 − 0.843i)6-s + (0.702 + 0.188i)7-s + (0.585 + 0.585i)8-s + (−0.992 − 0.125i)9-s + (−0.828 + 0.478i)10-s + (−0.369 − 0.369i)11-s + (−0.263 − 0.0165i)12-s + (0.731 + 0.682i)13-s + (0.708 − 0.408i)14-s + (−0.271 + 0.806i)15-s + 1.19·16-s + (−0.666 + 1.15i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.0346+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0346 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.0346+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0346 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.0346+0.999i0.0346 + 0.999i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(11,⋅)\chi_{117} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.0346+0.999i)(2,\ 117,\ (\ :1/2),\ 0.0346 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.04296−1.00743i1.04296 - 1.00743i
L(12)L(\frac12) ≈\approx 1.04296−1.00743i1.04296 - 1.00743i
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+(−0.108+1.72i)T 1 + (-0.108 + 1.72i)T
13 1+(−2.63−2.46i)T 1 + (-2.63 - 2.46i)T
good2 1+(−1.12+1.12i)T−2iT2 1 + (-1.12 + 1.12i)T - 2iT^{2}
5 1+(1.83+0.492i)T+(4.33+2.5i)T2 1 + (1.83 + 0.492i)T + (4.33 + 2.5i)T^{2}
7 1+(−1.85−0.498i)T+(6.06+3.5i)T2 1 + (-1.85 - 0.498i)T + (6.06 + 3.5i)T^{2}
11 1+(1.22+1.22i)T+11iT2 1 + (1.22 + 1.22i)T + 11iT^{2}
17 1+(2.74−4.76i)T+(−8.5−14.7i)T2 1 + (2.74 - 4.76i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.121−0.0326i)T+(16.4−9.5i)T2 1 + (0.121 - 0.0326i)T + (16.4 - 9.5i)T^{2}
23 1+(−2.99+5.18i)T+(−11.5−19.9i)T2 1 + (-2.99 + 5.18i)T + (-11.5 - 19.9i)T^{2}
29 1+4.13iT−29T2 1 + 4.13iT - 29T^{2}
31 1+(1.98−7.42i)T+(−26.8−15.5i)T2 1 + (1.98 - 7.42i)T + (-26.8 - 15.5i)T^{2}
37 1+(3.76+1.00i)T+(32.0+18.5i)T2 1 + (3.76 + 1.00i)T + (32.0 + 18.5i)T^{2}
41 1+(−0.0773−0.288i)T+(−35.5+20.5i)T2 1 + (-0.0773 - 0.288i)T + (-35.5 + 20.5i)T^{2}
43 1+(1.76−1.01i)T+(21.5−37.2i)T2 1 + (1.76 - 1.01i)T + (21.5 - 37.2i)T^{2}
47 1+(10.4−2.81i)T+(40.7−23.5i)T2 1 + (10.4 - 2.81i)T + (40.7 - 23.5i)T^{2}
53 1+9.44iT−53T2 1 + 9.44iT - 53T^{2}
59 1+(−5.60−5.60i)T+59iT2 1 + (-5.60 - 5.60i)T + 59iT^{2}
61 1+(6.37+11.0i)T+(−30.5+52.8i)T2 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(−11.8+3.17i)T+(58.0−33.5i)T2 1 + (-11.8 + 3.17i)T + (58.0 - 33.5i)T^{2}
71 1+(−1.16−4.34i)T+(−61.4+35.5i)T2 1 + (-1.16 - 4.34i)T + (-61.4 + 35.5i)T^{2}
73 1+(−9.98+9.98i)T−73iT2 1 + (-9.98 + 9.98i)T - 73iT^{2}
79 1+(−1.34+2.33i)T+(−39.5−68.4i)T2 1 + (-1.34 + 2.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(0.0823+0.307i)T+(−71.8+41.5i)T2 1 + (0.0823 + 0.307i)T + (-71.8 + 41.5i)T^{2}
89 1+(2.68−10.0i)T+(−77.0−44.5i)T2 1 + (2.68 - 10.0i)T + (-77.0 - 44.5i)T^{2}
97 1+(−1.02+3.83i)T+(−84.0−48.5i)T2 1 + (-1.02 + 3.83i)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

−13.05234898726485966895476909876, −12.34443381477391253800831585842, −11.42341627589721130453655451852, −10.88841883855551029396264429948, −8.503285116265374558427734049094, −8.094243110090268608863382526672, −6.53490196154291846211244518272, −4.92119381911101215061081483124, −3.58168841235858272505351994959, −1.91938954197979256541738388742, 3.52001748734375843034002917630, 4.66557750396078821159613933064, 5.53698421328843441778742689605, 7.15755179291424844946896758325, 8.145237537244317860592599502222, 9.604383071186482842420944455450, 10.84587713094759731448348927488, 11.51225933336017015570799329793, 13.15596142491587332674548900801, 14.04272156042335486695147202481

Graph of the ZZ-function along the critical line