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Properties

Label 2-370-37.25-c1-0-5
Degree 22
Conductor 370370
Sign 0.984+0.173i0.984 + 0.173i
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  + (0.642 + 0.766i)2-s + (−1.89 − 1.58i)3-s + (−0.173 + 0.984i)4-s + (0.342 + 0.939i)5-s − 2.46i·6-s + (2.50 − 0.911i)7-s + (−0.866 + 0.500i)8-s + (0.537 + 3.04i)9-s + (−0.5 + 0.866i)10-s + (−1.28 − 2.23i)11-s + (1.89 − 1.58i)12-s + (2.22 + 0.392i)13-s + (2.30 + 1.33i)14-s + (0.844 − 2.31i)15-s + (−0.939 − 0.342i)16-s + (5.64 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−1.09 − 0.916i)3-s + (−0.0868 + 0.492i)4-s + (0.152 + 0.420i)5-s − 1.00i·6-s + (0.946 − 0.344i)7-s + (−0.306 + 0.176i)8-s + (0.179 + 1.01i)9-s + (−0.158 + 0.273i)10-s + (−0.388 − 0.672i)11-s + (0.545 − 0.458i)12-s + (0.617 + 0.108i)13-s + (0.616 + 0.356i)14-s + (0.217 − 0.598i)15-s + (−0.234 − 0.0855i)16-s + (1.36 − 0.241i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.984+0.173i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.984+0.173i0.984 + 0.173i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(321,⋅)\chi_{370} (321, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.984+0.173i)(2,\ 370,\ (\ :1/2),\ 0.984 + 0.173i)

Particular Values

L(1)L(1) ≈\approx 1.35800−0.118739i1.35800 - 0.118739i
L(12)L(\frac12) ≈\approx 1.35800−0.118739i1.35800 - 0.118739i
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+(−0.642−0.766i)T 1 + (-0.642 - 0.766i)T
5 1+(−0.342−0.939i)T 1 + (-0.342 - 0.939i)T
37 1+(−0.874−6.01i)T 1 + (-0.874 - 6.01i)T
good3 1+(1.89+1.58i)T+(0.520+2.95i)T2 1 + (1.89 + 1.58i)T + (0.520 + 2.95i)T^{2}
7 1+(−2.50+0.911i)T+(5.36−4.49i)T2 1 + (-2.50 + 0.911i)T + (5.36 - 4.49i)T^{2}
11 1+(1.28+2.23i)T+(−5.5+9.52i)T2 1 + (1.28 + 2.23i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.22−0.392i)T+(12.2+4.44i)T2 1 + (-2.22 - 0.392i)T + (12.2 + 4.44i)T^{2}
17 1+(−5.64+0.995i)T+(15.9−5.81i)T2 1 + (-5.64 + 0.995i)T + (15.9 - 5.81i)T^{2}
19 1+(−3.50+4.17i)T+(−3.29−18.7i)T2 1 + (-3.50 + 4.17i)T + (-3.29 - 18.7i)T^{2}
23 1+(−4.45−2.57i)T+(11.5+19.9i)T2 1 + (-4.45 - 2.57i)T + (11.5 + 19.9i)T^{2}
29 1+(−1.31+0.757i)T+(14.5−25.1i)T2 1 + (-1.31 + 0.757i)T + (14.5 - 25.1i)T^{2}
31 1+9.01iT−31T2 1 + 9.01iT - 31T^{2}
41 1+(0.500−2.83i)T+(−38.5−14.0i)T2 1 + (0.500 - 2.83i)T + (-38.5 - 14.0i)T^{2}
43 1+4.92iT−43T2 1 + 4.92iT - 43T^{2}
47 1+(5.13−8.88i)T+(−23.5−40.7i)T2 1 + (5.13 - 8.88i)T + (-23.5 - 40.7i)T^{2}
53 1+(10.2+3.72i)T+(40.6+34.0i)T2 1 + (10.2 + 3.72i)T + (40.6 + 34.0i)T^{2}
59 1+(0.985−2.70i)T+(−45.1−37.9i)T2 1 + (0.985 - 2.70i)T + (-45.1 - 37.9i)T^{2}
61 1+(8.49+1.49i)T+(57.3+20.8i)T2 1 + (8.49 + 1.49i)T + (57.3 + 20.8i)T^{2}
67 1+(0.287−0.104i)T+(51.3−43.0i)T2 1 + (0.287 - 0.104i)T + (51.3 - 43.0i)T^{2}
71 1+(3.30+2.77i)T+(12.3+69.9i)T2 1 + (3.30 + 2.77i)T + (12.3 + 69.9i)T^{2}
73 1−2.58T+73T2 1 - 2.58T + 73T^{2}
79 1+(−3.01−8.28i)T+(−60.5+50.7i)T2 1 + (-3.01 - 8.28i)T + (-60.5 + 50.7i)T^{2}
83 1+(−0.982−5.57i)T+(−77.9+28.3i)T2 1 + (-0.982 - 5.57i)T + (-77.9 + 28.3i)T^{2}
89 1+(4.18−11.4i)T+(−68.1−57.2i)T2 1 + (4.18 - 11.4i)T + (-68.1 - 57.2i)T^{2}
97 1+(−12.7−7.38i)T+(48.5+84.0i)T2 1 + (-12.7 - 7.38i)T + (48.5 + 84.0i)T^{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

−11.27686704409387808662934691087, −11.08966033279797379401586592453, −9.549673146426650167878569391221, −8.034577410694112233259722377844, −7.48645307948768043563038789665, −6.48188289016457929769952186142, −5.66129437742363820687891123139, −4.85689670168182003072064548824, −3.15977368784875660623299049689, −1.15026135678703671053536338190, 1.45916777376060705770397715150, 3.43885789833133695254362011277, 4.79421635437769374818552421796, 5.19178079403347332977954137278, 6.09009172323459240331366519103, 7.72730372004526148120791012744, 8.909084815034889076196991544730, 10.05020446112988280212389503195, 10.54071896299092404500637926038, 11.43952723195964215830781836666

Graph of the ZZ-function along the critical line