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Properties

Label 2-392-56.27-c1-0-31
Degree 22
Conductor 392392
Sign −0.156+0.987i-0.156 + 0.987i
Analytic cond. 3.130133.13013
Root an. cond. 1.769211.76921
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  + 1.41·2-s − 3.37i·3-s + 2.00·4-s − 4.77i·6-s + 2.82·8-s − 8.41·9-s + 2.24·11-s − 6.75i·12-s + 4.00·16-s + 3.37i·17-s − 11.8·18-s + 1.39i·19-s + 3.17·22-s − 9.55i·24-s − 5·25-s + ⋯
L(s)  = 1  + 1.00·2-s − 1.95i·3-s + 1.00·4-s − 1.95i·6-s + 1.00·8-s − 2.80·9-s + 0.676·11-s − 1.95i·12-s + 1.00·16-s + 0.819i·17-s − 2.80·18-s + 0.321i·19-s + 0.676·22-s − 1.95i·24-s − 25-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((−0.156+0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+1/2) L(s)=((−0.156+0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: −0.156+0.987i-0.156 + 0.987i
Analytic conductor: 3.130133.13013
Root analytic conductor: 1.769211.76921
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ392(195,⋅)\chi_{392} (195, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 392, ( :1/2), −0.156+0.987i)(2,\ 392,\ (\ :1/2),\ -0.156 + 0.987i)

Particular Values

L(1)L(1) ≈\approx 1.58950−1.86131i1.58950 - 1.86131i
L(12)L(\frac12) ≈\approx 1.58950−1.86131i1.58950 - 1.86131i
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 1−1.41T 1 - 1.41T
7 1 1
good3 1+3.37iT−3T2 1 + 3.37iT - 3T^{2}
5 1+5T2 1 + 5T^{2}
11 1−2.24T+11T2 1 - 2.24T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−3.37iT−17T2 1 - 3.37iT - 17T^{2}
19 1−1.39iT−19T2 1 - 1.39iT - 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+8.15iT−41T2 1 + 8.15iT - 41T^{2}
43 1−13.0T+43T2 1 - 13.0T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−10.9iT−59T2 1 - 10.9iT - 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+8.48T+67T2 1 + 8.48T + 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1−14.9iT−73T2 1 - 14.9iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+17.7iT−83T2 1 + 17.7iT - 83T^{2}
89 1+12.1iT−89T2 1 + 12.1iT - 89T^{2}
97 1−15.7iT−97T2 1 - 15.7iT - 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

−11.59296748127714518510175290143, −10.59040095074913453359037541923, −8.886687424290441428635575306380, −7.82093062013373173431685175971, −7.17152183556223483653360433100, −6.20494577969483761611031721415, −5.66146869234467549316520572774, −3.89467172415305145333909008026, −2.49852174616488983608155380727, −1.42251241089855163724047909662, 2.73370984822445473800108118628, 3.78752791798539252294808926140, 4.54746904003177883314268759710, 5.42634059451941202387515861568, 6.37675623192398542502351789208, 7.87610034453133981741459986903, 9.171233930762076984237980125857, 9.811952300313011638445835120686, 10.83414947562781700621008444335, 11.40433405515417077862465219235

Graph of the ZZ-function along the critical line