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Properties

Label 2-370-185.162-c1-0-7
Degree 22
Conductor 370370
Sign 0.866−0.499i0.866 - 0.499i
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  + (−0.866 − 0.5i)2-s + (−0.392 + 1.46i)3-s + (0.499 + 0.866i)4-s + (2.15 − 0.582i)5-s + (1.07 − 1.07i)6-s + (−0.155 + 0.578i)7-s − 0.999i·8-s + (0.603 + 0.348i)9-s + (−2.16 − 0.575i)10-s − 2.03i·11-s + (−1.46 + 0.392i)12-s + (3.91 − 2.26i)13-s + (0.423 − 0.423i)14-s + (0.00519 + 3.39i)15-s + (−0.5 + 0.866i)16-s + (−3.22 + 5.58i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.226 + 0.846i)3-s + (0.249 + 0.433i)4-s + (0.965 − 0.260i)5-s + (0.438 − 0.438i)6-s + (−0.0586 + 0.218i)7-s − 0.353i·8-s + (0.201 + 0.116i)9-s + (−0.683 − 0.181i)10-s − 0.613i·11-s + (−0.423 + 0.113i)12-s + (1.08 − 0.626i)13-s + (0.113 − 0.113i)14-s + (0.00134 + 0.876i)15-s + (−0.125 + 0.216i)16-s + (−0.782 + 1.35i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.866−0.499i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.866−0.499i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.866−0.499i0.866 - 0.499i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(347,⋅)\chi_{370} (347, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.866−0.499i)(2,\ 370,\ (\ :1/2),\ 0.866 - 0.499i)

Particular Values

L(1)L(1) ≈\approx 1.12072+0.299925i1.12072 + 0.299925i
L(12)L(\frac12) ≈\approx 1.12072+0.299925i1.12072 + 0.299925i
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
5 1+(−2.15+0.582i)T 1 + (-2.15 + 0.582i)T
37 1+(0.586−6.05i)T 1 + (0.586 - 6.05i)T
good3 1+(0.392−1.46i)T+(−2.59−1.5i)T2 1 + (0.392 - 1.46i)T + (-2.59 - 1.5i)T^{2}
7 1+(0.155−0.578i)T+(−6.06−3.5i)T2 1 + (0.155 - 0.578i)T + (-6.06 - 3.5i)T^{2}
11 1+2.03iT−11T2 1 + 2.03iT - 11T^{2}
13 1+(−3.91+2.26i)T+(6.5−11.2i)T2 1 + (-3.91 + 2.26i)T + (6.5 - 11.2i)T^{2}
17 1+(3.22−5.58i)T+(−8.5−14.7i)T2 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.311+1.16i)T+(−16.4−9.5i)T2 1 + (-0.311 + 1.16i)T + (-16.4 - 9.5i)T^{2}
23 1−4.39iT−23T2 1 - 4.39iT - 23T^{2}
29 1+(−5.43+5.43i)T−29iT2 1 + (-5.43 + 5.43i)T - 29iT^{2}
31 1+(−5.48−5.48i)T+31iT2 1 + (-5.48 - 5.48i)T + 31iT^{2}
41 1+(1.10−0.637i)T+(20.5−35.5i)T2 1 + (1.10 - 0.637i)T + (20.5 - 35.5i)T^{2}
43 1+3.83iT−43T2 1 + 3.83iT - 43T^{2}
47 1+(3.13+3.13i)T+47iT2 1 + (3.13 + 3.13i)T + 47iT^{2}
53 1+(−0.0992−0.370i)T+(−45.8+26.5i)T2 1 + (-0.0992 - 0.370i)T + (-45.8 + 26.5i)T^{2}
59 1+(10.2−2.74i)T+(51.0−29.5i)T2 1 + (10.2 - 2.74i)T + (51.0 - 29.5i)T^{2}
61 1+(−2.46+9.19i)T+(−52.8−30.5i)T2 1 + (-2.46 + 9.19i)T + (-52.8 - 30.5i)T^{2}
67 1+(0.762+0.204i)T+(58.0+33.5i)T2 1 + (0.762 + 0.204i)T + (58.0 + 33.5i)T^{2}
71 1+(8.36+14.4i)T+(−35.5+61.4i)T2 1 + (8.36 + 14.4i)T + (-35.5 + 61.4i)T^{2}
73 1+(3.81+3.81i)T+73iT2 1 + (3.81 + 3.81i)T + 73iT^{2}
79 1+(3.16−11.8i)T+(−68.4−39.5i)T2 1 + (3.16 - 11.8i)T + (-68.4 - 39.5i)T^{2}
83 1+(−0.261−0.974i)T+(−71.8+41.5i)T2 1 + (-0.261 - 0.974i)T + (-71.8 + 41.5i)T^{2}
89 1+(2.11+7.88i)T+(−77.0+44.5i)T2 1 + (2.11 + 7.88i)T + (-77.0 + 44.5i)T^{2}
97 1+15.7T+97T2 1 + 15.7T + 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

−11.03980624239536191493509327200, −10.48994812804442072418717643821, −9.809684698718677408322655539572, −8.824738414151242583639113290228, −8.196118789434269480066457389312, −6.52235634787424048203468229067, −5.69830625953032433156244366103, −4.45637739941817032854969133438, −3.15497929333184623340605459469, −1.51951803219766543862604745407, 1.20655725215176595612969497953, 2.46414236861265247603398714849, 4.52251403219934752794624015610, 5.96731499121195121756077058981, 6.71266378779063793661956644507, 7.25286968081813085529163602062, 8.592793176099152788864117259841, 9.450018023096190922615914325846, 10.27024660714032406195604013966, 11.22674815534194572226889783365

Graph of the ZZ-function along the critical line