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Properties

Label 2-760-5.4-c1-0-10
Degree 22
Conductor 760760
Sign 0.698−0.715i0.698 - 0.715i
Analytic cond. 6.068636.06863
Root an. cond. 2.463452.46345
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  + 1.59i·3-s + (−1.59 − 1.56i)5-s + 1.66i·7-s + 0.463·9-s + 5.56·11-s − 6.31i·13-s + (2.48 − 2.54i)15-s + 4.12i·17-s − 19-s − 2.64·21-s + 1.82i·23-s + (0.114 + 4.99i)25-s + 5.51i·27-s + 4.08·29-s + 6.61·31-s + ⋯
L(s)  = 1  + 0.919i·3-s + (−0.715 − 0.698i)5-s + 0.628i·7-s + 0.154·9-s + 1.67·11-s − 1.75i·13-s + (0.642 − 0.657i)15-s + 0.999i·17-s − 0.229·19-s − 0.577·21-s + 0.380i·23-s + (0.0228 + 0.999i)25-s + 1.06i·27-s + 0.759·29-s + 1.18·31-s + ⋯

Functional equation

Λ(s)=(760s/2 ΓC(s) L(s)=((0.698−0.715i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(760s/2 ΓC(s+1/2) L(s)=((0.698−0.715i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 760760    =    23⋅5⋅192^{3} \cdot 5 \cdot 19
Sign: 0.698−0.715i0.698 - 0.715i
Analytic conductor: 6.068636.06863
Root analytic conductor: 2.463452.46345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ760(609,⋅)\chi_{760} (609, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 760, ( :1/2), 0.698−0.715i)(2,\ 760,\ (\ :1/2),\ 0.698 - 0.715i)

Particular Values

L(1)L(1) ≈\approx 1.40462+0.591230i1.40462 + 0.591230i
L(12)L(\frac12) ≈\approx 1.40462+0.591230i1.40462 + 0.591230i
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)
bad2 1 1
5 1+(1.59+1.56i)T 1 + (1.59 + 1.56i)T
19 1+T 1 + T
good3 1−1.59iT−3T2 1 - 1.59iT - 3T^{2}
7 1−1.66iT−7T2 1 - 1.66iT - 7T^{2}
11 1−5.56T+11T2 1 - 5.56T + 11T^{2}
13 1+6.31iT−13T2 1 + 6.31iT - 13T^{2}
17 1−4.12iT−17T2 1 - 4.12iT - 17T^{2}
23 1−1.82iT−23T2 1 - 1.82iT - 23T^{2}
29 1−4.08T+29T2 1 - 4.08T + 29T^{2}
31 1−6.61T+31T2 1 - 6.61T + 31T^{2}
37 1−9.66iT−37T2 1 - 9.66iT - 37T^{2}
41 1+4.61T+41T2 1 + 4.61T + 41T^{2}
43 1−3.75iT−43T2 1 - 3.75iT - 43T^{2}
47 1+3.85iT−47T2 1 + 3.85iT - 47T^{2}
53 1+5.24iT−53T2 1 + 5.24iT - 53T^{2}
59 1−11.5T+59T2 1 - 11.5T + 59T^{2}
61 1−8.02T+61T2 1 - 8.02T + 61T^{2}
67 1−0.155iT−67T2 1 - 0.155iT - 67T^{2}
71 1+12.1T+71T2 1 + 12.1T + 71T^{2}
73 1+0.795iT−73T2 1 + 0.795iT - 73T^{2}
79 1−7.18T+79T2 1 - 7.18T + 79T^{2}
83 1+5.88iT−83T2 1 + 5.88iT - 83T^{2}
89 1+18.5T+89T2 1 + 18.5T + 89T^{2}
97 1−2.77iT−97T2 1 - 2.77iT - 97T^{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

−10.22892243132131939814432430363, −9.720932829680840335506758582726, −8.529180525645778800779709902434, −8.360445848053891872653996964625, −6.93908069361247496266239138460, −5.84610157080938846926080604428, −4.88249879759487678289728892353, −4.04116832240462592217714432245, −3.22714023704461296145597557069, −1.23010730108682489659018207044, 1.01744621884474014362479267323, 2.34828176369544434625206606311, 3.91601094573919204294430901812, 4.40408729270776271137497397592, 6.33618770564950780982799802877, 6.93300309704741332280698591749, 7.21694408319060245980913450831, 8.434641436070484929294332894114, 9.316965018973412042982228443894, 10.26537963225840132677524599825

Graph of the ZZ-function along the critical line