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Properties

Label 2-370-37.36-c1-0-3
Degree 22
Conductor 370370
Sign 0.164−0.986i0.164 - 0.986i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  + i·2-s − 4-s + i·5-s + 5·7-s − i·8-s − 3·9-s − 10-s + 3·11-s + 2i·13-s + 5i·14-s + 16-s − i·17-s − 3i·18-s + 2i·19-s − i·20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 1.88·7-s − 0.353i·8-s − 9-s − 0.316·10-s + 0.904·11-s + 0.554i·13-s + 1.33i·14-s + 0.250·16-s − 0.242i·17-s − 0.707i·18-s + 0.458i·19-s − 0.223i·20-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.164−0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.164−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.164−0.986i0.164 - 0.986i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(221,⋅)\chi_{370} (221, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.164−0.986i)(2,\ 370,\ (\ :1/2),\ 0.164 - 0.986i)

Particular Values

L(1)L(1) ≈\approx 1.12137+0.949949i1.12137 + 0.949949i
L(12)L(\frac12) ≈\approx 1.12137+0.949949i1.12137 + 0.949949i
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−iT 1 - iT
5 1−iT 1 - iT
37 1+(−1+6i)T 1 + (-1 + 6i)T
good3 1+3T2 1 + 3T^{2}
7 1−5T+7T2 1 - 5T + 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
19 1−2iT−19T2 1 - 2iT - 19T^{2}
23 1−6iT−23T2 1 - 6iT - 23T^{2}
29 1−5iT−29T2 1 - 5iT - 29T^{2}
31 1−iT−31T2 1 - iT - 31T^{2}
41 1−5T+41T2 1 - 5T + 41T^{2}
43 1+11iT−43T2 1 + 11iT - 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+9T+53T2 1 + 9T + 53T^{2}
59 1+12iT−59T2 1 + 12iT - 59T^{2}
61 1+7iT−61T2 1 + 7iT - 61T^{2}
67 1−2T+67T2 1 - 2T + 67T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1−4iT−89T2 1 - 4iT - 89T^{2}
97 1+11iT−97T2 1 + 11iT - 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

−11.38535450603358434949651587528, −11.01813855821274131465918536013, −9.502988323329158951080709023435, −8.638622250309714165735668203535, −7.86666365125470232879567897046, −6.95509674041225330471271564641, −5.75614913427035035115577633899, −4.91001198159744368286971336078, −3.66799652141793241573627985898, −1.76595702036829419059033839363, 1.20415248561010885495150745839, 2.60439181039286588324359421768, 4.26241489100573263738454712543, 5.00510690401806604401580998990, 6.15982169094809552272739788908, 7.920384719287853773541256781167, 8.403454162043797059519129602577, 9.279632933407168197259545241829, 10.53704253329823385867232383739, 11.42731315511054187933035705937

Graph of the ZZ-function along the critical line