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Properties

Label 2-675-1.1-c1-0-19
Degree 22
Conductor 675675
Sign 11
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + 4.85·4-s + 7.47·8-s + 9.85·16-s + 3.76·17-s − 8.70·19-s − 1.47·23-s + 2.70·31-s + 10.8·32-s + 9.85·34-s − 22.7·38-s − 3.85·46-s − 8.94·47-s − 7·49-s + 14.2·53-s − 14.4·61-s + 7.09·62-s + 8.70·64-s + 18.2·68-s − 42.2·76-s − 14.7·79-s + 11.9·83-s − 7.14·92-s − 23.4·94-s − 18.3·98-s + 37.2·106-s + 17.8·107-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s + 2.64·8-s + 2.46·16-s + 0.912·17-s − 1.99·19-s − 0.306·23-s + 0.486·31-s + 1.91·32-s + 1.68·34-s − 3.69·38-s − 0.568·46-s − 1.30·47-s − 49-s + 1.95·53-s − 1.84·61-s + 0.900·62-s + 1.08·64-s + 2.21·68-s − 4.84·76-s − 1.65·79-s + 1.31·83-s − 0.745·92-s − 2.41·94-s − 1.85·98-s + 3.62·106-s + 1.72·107-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 1)(2,\ 675,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4003900704.400390070
L(12)L(\frac12) ≈\approx 4.4003900704.400390070
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
5 1 1
good2 1−2.61T+2T2 1 - 2.61T + 2T^{2}
7 1+7T2 1 + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−3.76T+17T2 1 - 3.76T + 17T^{2}
19 1+8.70T+19T2 1 + 8.70T + 19T^{2}
23 1+1.47T+23T2 1 + 1.47T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−2.70T+31T2 1 - 2.70T + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+8.94T+47T2 1 + 8.94T + 47T^{2}
53 1−14.2T+53T2 1 - 14.2T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+14.4T+61T2 1 + 14.4T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+14.7T+79T2 1 + 14.7T + 79T^{2}
83 1−11.9T+83T2 1 - 11.9T + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 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

−10.78038521942540907100259045443, −10.03621504316639370617559622912, −8.558508402508444795291817557582, −7.54860608131123895356374003622, −6.53137444575247663719978475609, −5.92123140089364848706671948637, −4.87442971353380959391737872935, −4.08807451385268501567567573099, −3.08045219779734005587003432611, −1.92860018740434397351518724248, 1.92860018740434397351518724248, 3.08045219779734005587003432611, 4.08807451385268501567567573099, 4.87442971353380959391737872935, 5.92123140089364848706671948637, 6.53137444575247663719978475609, 7.54860608131123895356374003622, 8.558508402508444795291817557582, 10.03621504316639370617559622912, 10.78038521942540907100259045443

Graph of the ZZ-function along the critical line