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Properties

Label 2-117-117.11-c1-0-4
Degree 22
Conductor 117117
Sign 0.211−0.977i0.211 - 0.977i
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  + (−0.662 + 0.662i)2-s + (1.56 + 0.749i)3-s + 1.12i·4-s + (0.101 + 0.0271i)5-s + (−1.53 + 0.538i)6-s + (−0.353 − 0.0946i)7-s + (−2.06 − 2.06i)8-s + (1.87 + 2.34i)9-s + (−0.0852 + 0.0492i)10-s + (−2.25 − 2.25i)11-s + (−0.840 + 1.75i)12-s + (3.51 + 0.821i)13-s + (0.296 − 0.171i)14-s + (0.137 + 0.118i)15-s + 0.499·16-s + (−0.713 + 1.23i)17-s + ⋯
L(s)  = 1  + (−0.468 + 0.468i)2-s + (0.901 + 0.432i)3-s + 0.560i·4-s + (0.0453 + 0.0121i)5-s + (−0.625 + 0.219i)6-s + (−0.133 − 0.0357i)7-s + (−0.731 − 0.731i)8-s + (0.625 + 0.780i)9-s + (−0.0269 + 0.0155i)10-s + (−0.678 − 0.678i)11-s + (−0.242 + 0.505i)12-s + (0.973 + 0.227i)13-s + (0.0793 − 0.0457i)14-s + (0.0356 + 0.0305i)15-s + 0.124·16-s + (−0.173 + 0.299i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.211−0.977i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.211−0.977i0.211 - 0.977i
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), 0.211−0.977i)(2,\ 117,\ (\ :1/2),\ 0.211 - 0.977i)

Particular Values

L(1)L(1) ≈\approx 0.823874+0.664515i0.823874 + 0.664515i
L(12)L(\frac12) ≈\approx 0.823874+0.664515i0.823874 + 0.664515i
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+(−1.56−0.749i)T 1 + (-1.56 - 0.749i)T
13 1+(−3.51−0.821i)T 1 + (-3.51 - 0.821i)T
good2 1+(0.662−0.662i)T−2iT2 1 + (0.662 - 0.662i)T - 2iT^{2}
5 1+(−0.101−0.0271i)T+(4.33+2.5i)T2 1 + (-0.101 - 0.0271i)T + (4.33 + 2.5i)T^{2}
7 1+(0.353+0.0946i)T+(6.06+3.5i)T2 1 + (0.353 + 0.0946i)T + (6.06 + 3.5i)T^{2}
11 1+(2.25+2.25i)T+11iT2 1 + (2.25 + 2.25i)T + 11iT^{2}
17 1+(0.713−1.23i)T+(−8.5−14.7i)T2 1 + (0.713 - 1.23i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.92+0.784i)T+(16.4−9.5i)T2 1 + (-2.92 + 0.784i)T + (16.4 - 9.5i)T^{2}
23 1+(−1.83+3.18i)T+(−11.5−19.9i)T2 1 + (-1.83 + 3.18i)T + (-11.5 - 19.9i)T^{2}
29 1+5.17iT−29T2 1 + 5.17iT - 29T^{2}
31 1+(−1.46+5.46i)T+(−26.8−15.5i)T2 1 + (-1.46 + 5.46i)T + (-26.8 - 15.5i)T^{2}
37 1+(4.32+1.15i)T+(32.0+18.5i)T2 1 + (4.32 + 1.15i)T + (32.0 + 18.5i)T^{2}
41 1+(−1.76−6.57i)T+(−35.5+20.5i)T2 1 + (-1.76 - 6.57i)T + (-35.5 + 20.5i)T^{2}
43 1+(−1.94+1.12i)T+(21.5−37.2i)T2 1 + (-1.94 + 1.12i)T + (21.5 - 37.2i)T^{2}
47 1+(4.42−1.18i)T+(40.7−23.5i)T2 1 + (4.42 - 1.18i)T + (40.7 - 23.5i)T^{2}
53 1−13.0iT−53T2 1 - 13.0iT - 53T^{2}
59 1+(−2.44−2.44i)T+59iT2 1 + (-2.44 - 2.44i)T + 59iT^{2}
61 1+(−3.12−5.41i)T+(−30.5+52.8i)T2 1 + (-3.12 - 5.41i)T + (-30.5 + 52.8i)T^{2}
67 1+(14.8−3.96i)T+(58.0−33.5i)T2 1 + (14.8 - 3.96i)T + (58.0 - 33.5i)T^{2}
71 1+(−1.56−5.85i)T+(−61.4+35.5i)T2 1 + (-1.56 - 5.85i)T + (-61.4 + 35.5i)T^{2}
73 1+(4.13−4.13i)T−73iT2 1 + (4.13 - 4.13i)T - 73iT^{2}
79 1+(−8.15+14.1i)T+(−39.5−68.4i)T2 1 + (-8.15 + 14.1i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.29+16.0i)T+(−71.8+41.5i)T2 1 + (4.29 + 16.0i)T + (-71.8 + 41.5i)T^{2}
89 1+(−0.630+2.35i)T+(−77.0−44.5i)T2 1 + (-0.630 + 2.35i)T + (-77.0 - 44.5i)T^{2}
97 1+(3.20−11.9i)T+(−84.0−48.5i)T2 1 + (3.20 - 11.9i)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

−13.62152925644921572891489284725, −13.12676230543906852521079970989, −11.62594151259679683072796938259, −10.34439429619991252067530937575, −9.221276917480413222463925729587, −8.369392802894357188810535220983, −7.59761945380543508043011898062, −6.12836780333993601457143143850, −4.15410005798290875439273172011, −2.89620233950953708193145692603, 1.65969914695201600667361128299, 3.24684859712359366844805849018, 5.32212392633173717968976939844, 6.84152733280874629132171007021, 8.119376900506964599941216339085, 9.176841166479062838011642880004, 9.976871801077293650207050038251, 11.08151574964284560613213791209, 12.31339220339387087354468075913, 13.39369475988908458893325690301

Graph of the ZZ-function along the critical line