Properties

Label 2-152-152.75-c1-0-8
Degree 22
Conductor 152152
Sign 0.9990.00742i0.999 - 0.00742i
Analytic cond. 1.213721.21372
Root an. cond. 1.101691.10169
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  + (1.17 − 0.792i)2-s + 1.26i·3-s + (0.743 − 1.85i)4-s + 3.51i·5-s + (1 + 1.47i)6-s − 2.23i·7-s + (−0.601 − 2.76i)8-s + 1.40·9-s + (2.78 + 4.12i)10-s + 2.89·11-s + (2.34 + 0.937i)12-s − 6.30·13-s + (−1.77 − 2.61i)14-s − 4.43·15-s + (−2.89 − 2.76i)16-s − 4.79·17-s + ⋯
L(s)  = 1  + (0.828 − 0.560i)2-s + 0.728i·3-s + (0.371 − 0.928i)4-s + 1.57i·5-s + (0.408 + 0.603i)6-s − 0.845i·7-s + (−0.212 − 0.977i)8-s + 0.469·9-s + (0.882 + 1.30i)10-s + 0.872·11-s + (0.676 + 0.270i)12-s − 1.74·13-s + (−0.473 − 0.699i)14-s − 1.14·15-s + (−0.723 − 0.690i)16-s − 1.16·17-s + ⋯

Functional equation

Λ(s)=(152s/2ΓC(s)L(s)=((0.9990.00742i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00742i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2ΓC(s+1/2)L(s)=((0.9990.00742i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00742i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23192^{3} \cdot 19
Sign: 0.9990.00742i0.999 - 0.00742i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(75,)\chi_{152} (75, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), 0.9990.00742i)(2,\ 152,\ (\ :1/2),\ 0.999 - 0.00742i)

Particular Values

L(1)L(1) \approx 1.69259+0.00628203i1.69259 + 0.00628203i
L(12)L(\frac12) \approx 1.69259+0.00628203i1.69259 + 0.00628203i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.17+0.792i)T 1 + (-1.17 + 0.792i)T
19 1+(0.895+4.26i)T 1 + (-0.895 + 4.26i)T
good3 11.26iT3T2 1 - 1.26iT - 3T^{2}
5 13.51iT5T2 1 - 3.51iT - 5T^{2}
7 1+2.23iT7T2 1 + 2.23iT - 7T^{2}
11 12.89T+11T2 1 - 2.89T + 11T^{2}
13 1+6.30T+13T2 1 + 6.30T + 13T^{2}
17 1+4.79T+17T2 1 + 4.79T + 17T^{2}
23 1+0.524iT23T2 1 + 0.524iT - 23T^{2}
29 10.415T+29T2 1 - 0.415T + 29T^{2}
31 11.20T+31T2 1 - 1.20T + 31T^{2}
37 15.88T+37T2 1 - 5.88T + 37T^{2}
41 14.87iT41T2 1 - 4.87iT - 41T^{2}
43 1+10.6T+43T2 1 + 10.6T + 43T^{2}
47 1+4.27iT47T2 1 + 4.27iT - 47T^{2}
53 16.05T+53T2 1 - 6.05T + 53T^{2}
59 1+8.08iT59T2 1 + 8.08iT - 59T^{2}
61 18.38iT61T2 1 - 8.38iT - 61T^{2}
67 19.79iT67T2 1 - 9.79iT - 67T^{2}
71 110.2T+71T2 1 - 10.2T + 71T^{2}
73 17.76T+73T2 1 - 7.76T + 73T^{2}
79 1+7.33T+79T2 1 + 7.33T + 79T^{2}
83 12T+83T2 1 - 2T + 83T^{2}
89 112.1iT89T2 1 - 12.1iT - 89T^{2}
97 1+2.19iT97T2 1 + 2.19iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−13.14462850675538789688551154582, −11.74203368274968186935728059604, −10.99083382043444719162600933526, −10.15549384259909853588677396030, −9.560610797848217401175824163095, −7.09840289472504744007593242374, −6.72230089693255330090973792315, −4.80788758680666564587646636720, −3.87456427252729340573779285553, −2.56031094997546536249620810375, 2.05344631968229969942982782198, 4.30693357454120374388475122856, 5.22026986713213091880322157099, 6.45764449576966844874723350908, 7.61587160948123993200056348692, 8.616764249550777714323104156174, 9.582095765145775153227139495626, 11.79783632444781162566417866936, 12.27453423792831390391718966579, 12.83776245530726145310363963597

Graph of the ZZ-function along the critical line