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Properties

Label 2-115-5.2-c2-0-7
Degree 22
Conductor 115115
Sign 0.813+0.582i0.813 + 0.582i
Analytic cond. 3.133523.13352
Root an. cond. 1.770171.77017
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.95 − 1.95i)2-s + (−0.210 + 0.210i)3-s + 3.61i·4-s + (4.62 + 1.89i)5-s + 0.820·6-s + (2.37 + 2.37i)7-s + (−0.749 + 0.749i)8-s + 8.91i·9-s + (−5.31 − 12.7i)10-s + 12.6·11-s + (−0.760 − 0.760i)12-s + (10.8 − 10.8i)13-s − 9.25i·14-s + (−1.37 + 0.573i)15-s + 17.3·16-s + (−15.9 − 15.9i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.975i)2-s + (−0.0700 + 0.0700i)3-s + 0.904i·4-s + (0.925 + 0.379i)5-s + 0.136·6-s + (0.338 + 0.338i)7-s + (−0.0936 + 0.0936i)8-s + 0.990i·9-s + (−0.531 − 1.27i)10-s + 1.15·11-s + (−0.0633 − 0.0633i)12-s + (0.835 − 0.835i)13-s − 0.661i·14-s + (−0.0914 + 0.0382i)15-s + 1.08·16-s + (−0.940 − 0.940i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.813+0.582i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1) L(s)=((0.813+0.582i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.813+0.582i0.813 + 0.582i
Analytic conductor: 3.133523.13352
Root analytic conductor: 1.770171.77017
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ115(47,⋅)\chi_{115} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1), 0.813+0.582i)(2,\ 115,\ (\ :1),\ 0.813 + 0.582i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.956664−0.307203i0.956664 - 0.307203i
L(12)L(\frac12) ≈\approx 0.956664−0.307203i0.956664 - 0.307203i
L(2)L(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)
bad5 1+(−4.62−1.89i)T 1 + (-4.62 - 1.89i)T
23 1+(−3.39+3.39i)T 1 + (-3.39 + 3.39i)T
good2 1+(1.95+1.95i)T+4iT2 1 + (1.95 + 1.95i)T + 4iT^{2}
3 1+(0.210−0.210i)T−9iT2 1 + (0.210 - 0.210i)T - 9iT^{2}
7 1+(−2.37−2.37i)T+49iT2 1 + (-2.37 - 2.37i)T + 49iT^{2}
11 1−12.6T+121T2 1 - 12.6T + 121T^{2}
13 1+(−10.8+10.8i)T−169iT2 1 + (-10.8 + 10.8i)T - 169iT^{2}
17 1+(15.9+15.9i)T+289iT2 1 + (15.9 + 15.9i)T + 289iT^{2}
19 1+0.760iT−361T2 1 + 0.760iT - 361T^{2}
29 1−42.2iT−841T2 1 - 42.2iT - 841T^{2}
31 1−37.1T+961T2 1 - 37.1T + 961T^{2}
37 1+(−33.8−33.8i)T+1.36e3iT2 1 + (-33.8 - 33.8i)T + 1.36e3iT^{2}
41 1+33.6T+1.68e3T2 1 + 33.6T + 1.68e3T^{2}
43 1+(−5.27+5.27i)T−1.84e3iT2 1 + (-5.27 + 5.27i)T - 1.84e3iT^{2}
47 1+(57.2+57.2i)T+2.20e3iT2 1 + (57.2 + 57.2i)T + 2.20e3iT^{2}
53 1+(36.4−36.4i)T−2.80e3iT2 1 + (36.4 - 36.4i)T - 2.80e3iT^{2}
59 1+22.9iT−3.48e3T2 1 + 22.9iT - 3.48e3T^{2}
61 1+89.6T+3.72e3T2 1 + 89.6T + 3.72e3T^{2}
67 1+(−54.0−54.0i)T+4.48e3iT2 1 + (-54.0 - 54.0i)T + 4.48e3iT^{2}
71 1−67.9T+5.04e3T2 1 - 67.9T + 5.04e3T^{2}
73 1+(10.7−10.7i)T−5.32e3iT2 1 + (10.7 - 10.7i)T - 5.32e3iT^{2}
79 1+92.3iT−6.24e3T2 1 + 92.3iT - 6.24e3T^{2}
83 1+(−7.05+7.05i)T−6.88e3iT2 1 + (-7.05 + 7.05i)T - 6.88e3iT^{2}
89 1+27.4iT−7.92e3T2 1 + 27.4iT - 7.92e3T^{2}
97 1+(27.7+27.7i)T+9.40e3iT2 1 + (27.7 + 27.7i)T + 9.40e3iT^{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

−13.14546073110867058303256045391, −11.71667142597259528229618209247, −10.97845851629786848924560299417, −10.16281019502451573087521227197, −9.143034006398008175225180262071, −8.285267362075964106796574322573, −6.56726192663553969405050752666, −5.15355806885957031895922207546, −2.89494194080091155416385096272, −1.54379055910652790973886531815, 1.27220418352267477200377503412, 4.13046505971364195242113765929, 6.32379681899621687667010493947, 6.42758063035731230439970773257, 8.171698007528709734773593771653, 9.127299310137227944041397550907, 9.679357440438611901388209441351, 11.18806887185687531441196914404, 12.43423235933009774611417788551, 13.65370827974727382569665790912

Graph of the ZZ-function along the critical line