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Properties

Label 2-115-5.2-c2-0-8
Degree 22
Conductor 115115
Sign −0.819−0.573i-0.819 - 0.573i
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  + (2.29 + 2.29i)2-s + (−2.60 + 2.60i)3-s + 6.55i·4-s + (4.64 + 1.84i)5-s − 11.9·6-s + (−4.55 − 4.55i)7-s + (−5.88 + 5.88i)8-s − 4.54i·9-s + (6.42 + 14.9i)10-s + 0.500·11-s + (−17.0 − 17.0i)12-s + (6.17 − 6.17i)13-s − 20.9i·14-s + (−16.8 + 7.27i)15-s − 0.789·16-s + (1.49 + 1.49i)17-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + (−0.867 + 0.867i)3-s + 1.63i·4-s + (0.929 + 0.369i)5-s − 1.99·6-s + (−0.650 − 0.650i)7-s + (−0.735 + 0.735i)8-s − 0.504i·9-s + (0.642 + 1.49i)10-s + 0.0455·11-s + (−1.42 − 1.42i)12-s + (0.475 − 0.475i)13-s − 1.49i·14-s + (−1.12 + 0.485i)15-s − 0.0493·16-s + (0.0876 + 0.0876i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.819−0.573i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1) L(s)=((−0.819−0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.819−0.573i-0.819 - 0.573i
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), −0.819−0.573i)(2,\ 115,\ (\ :1),\ -0.819 - 0.573i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.605573+1.92156i0.605573 + 1.92156i
L(12)L(\frac12) ≈\approx 0.605573+1.92156i0.605573 + 1.92156i
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+(−4.64−1.84i)T 1 + (-4.64 - 1.84i)T
23 1+(3.39−3.39i)T 1 + (3.39 - 3.39i)T
good2 1+(−2.29−2.29i)T+4iT2 1 + (-2.29 - 2.29i)T + 4iT^{2}
3 1+(2.60−2.60i)T−9iT2 1 + (2.60 - 2.60i)T - 9iT^{2}
7 1+(4.55+4.55i)T+49iT2 1 + (4.55 + 4.55i)T + 49iT^{2}
11 1−0.500T+121T2 1 - 0.500T + 121T^{2}
13 1+(−6.17+6.17i)T−169iT2 1 + (-6.17 + 6.17i)T - 169iT^{2}
17 1+(−1.49−1.49i)T+289iT2 1 + (-1.49 - 1.49i)T + 289iT^{2}
19 1−21.0iT−361T2 1 - 21.0iT - 361T^{2}
29 1+43.1iT−841T2 1 + 43.1iT - 841T^{2}
31 1−57.0T+961T2 1 - 57.0T + 961T^{2}
37 1+(13.6+13.6i)T+1.36e3iT2 1 + (13.6 + 13.6i)T + 1.36e3iT^{2}
41 1−31.0T+1.68e3T2 1 - 31.0T + 1.68e3T^{2}
43 1+(40.9−40.9i)T−1.84e3iT2 1 + (40.9 - 40.9i)T - 1.84e3iT^{2}
47 1+(60.7+60.7i)T+2.20e3iT2 1 + (60.7 + 60.7i)T + 2.20e3iT^{2}
53 1+(−0.965+0.965i)T−2.80e3iT2 1 + (-0.965 + 0.965i)T - 2.80e3iT^{2}
59 1+1.59iT−3.48e3T2 1 + 1.59iT - 3.48e3T^{2}
61 1+14.5T+3.72e3T2 1 + 14.5T + 3.72e3T^{2}
67 1+(91.2+91.2i)T+4.48e3iT2 1 + (91.2 + 91.2i)T + 4.48e3iT^{2}
71 1+72.1T+5.04e3T2 1 + 72.1T + 5.04e3T^{2}
73 1+(−59.6+59.6i)T−5.32e3iT2 1 + (-59.6 + 59.6i)T - 5.32e3iT^{2}
79 1+80.4iT−6.24e3T2 1 + 80.4iT - 6.24e3T^{2}
83 1+(45.6−45.6i)T−6.88e3iT2 1 + (45.6 - 45.6i)T - 6.88e3iT^{2}
89 1−61.2iT−7.92e3T2 1 - 61.2iT - 7.92e3T^{2}
97 1+(−112.−112.i)T+9.40e3iT2 1 + (-112. - 112. i)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

−13.71752692587925909305624501341, −13.21811333466877564535935779028, −11.86747655485967929579170704089, −10.42398247536629833343811542820, −9.885821292772197936160154802992, −7.908748814163412429453999921872, −6.41416243908149227384811226263, −5.93283312396439288212737163977, −4.76675738186697047168328761327, −3.53937655759855208910174642658, 1.35214992675166087079776376684, 2.84002486938633519663327937026, 4.80972827599538769499566603189, 5.87797397489246305358604258043, 6.65635870945101280477359824442, 8.938028308152661614331890559628, 10.13273594269713157401044574120, 11.30406890598833589371230687248, 12.09177508792894422296116566224, 12.84091301731222931393417299294

Graph of the ZZ-function along the critical line