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Properties

Label 2-370-185.64-c1-0-17
Degree 22
Conductor 370370
Sign 0.0608+0.998i0.0608 + 0.998i
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  + (0.5 − 0.866i)2-s + (1.79 − 1.03i)3-s + (−0.499 − 0.866i)4-s + (1.99 − 1.00i)5-s − 2.06i·6-s + (1.25 − 0.725i)7-s − 0.999·8-s + (0.638 − 1.10i)9-s + (0.130 − 2.23i)10-s − 5.11·11-s + (−1.79 − 1.03i)12-s + (2.79 + 4.84i)13-s − 1.45i·14-s + (2.54 − 3.86i)15-s + (−0.5 + 0.866i)16-s + (−2.19 + 3.80i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (1.03 − 0.596i)3-s + (−0.249 − 0.433i)4-s + (0.893 − 0.448i)5-s − 0.844i·6-s + (0.474 − 0.274i)7-s − 0.353·8-s + (0.212 − 0.368i)9-s + (0.0413 − 0.705i)10-s − 1.54·11-s + (−0.516 − 0.298i)12-s + (0.775 + 1.34i)13-s − 0.387i·14-s + (0.656 − 0.997i)15-s + (−0.125 + 0.216i)16-s + (−0.532 + 0.921i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.0608+0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0608 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.0608+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0608 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.0608+0.998i0.0608 + 0.998i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(249,⋅)\chi_{370} (249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.0608+0.998i)(2,\ 370,\ (\ :1/2),\ 0.0608 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.71278−1.61157i1.71278 - 1.61157i
L(12)L(\frac12) ≈\approx 1.71278−1.61157i1.71278 - 1.61157i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1+(−1.99+1.00i)T 1 + (-1.99 + 1.00i)T
37 1+(−0.667+6.04i)T 1 + (-0.667 + 6.04i)T
good3 1+(−1.79+1.03i)T+(1.5−2.59i)T2 1 + (-1.79 + 1.03i)T + (1.5 - 2.59i)T^{2}
7 1+(−1.25+0.725i)T+(3.5−6.06i)T2 1 + (-1.25 + 0.725i)T + (3.5 - 6.06i)T^{2}
11 1+5.11T+11T2 1 + 5.11T + 11T^{2}
13 1+(−2.79−4.84i)T+(−6.5+11.2i)T2 1 + (-2.79 - 4.84i)T + (-6.5 + 11.2i)T^{2}
17 1+(2.19−3.80i)T+(−8.5−14.7i)T2 1 + (2.19 - 3.80i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.736−0.425i)T+(9.5−16.4i)T2 1 + (0.736 - 0.425i)T + (9.5 - 16.4i)T^{2}
23 1+1.02T+23T2 1 + 1.02T + 23T^{2}
29 1−3.13iT−29T2 1 - 3.13iT - 29T^{2}
31 1+6.98iT−31T2 1 + 6.98iT - 31T^{2}
41 1+(5.10+8.84i)T+(−20.5+35.5i)T2 1 + (5.10 + 8.84i)T + (-20.5 + 35.5i)T^{2}
43 1+3.37T+43T2 1 + 3.37T + 43T^{2}
47 1−3.87iT−47T2 1 - 3.87iT - 47T^{2}
53 1+(−9.19−5.30i)T+(26.5+45.8i)T2 1 + (-9.19 - 5.30i)T + (26.5 + 45.8i)T^{2}
59 1+(−1.97−1.14i)T+(29.5+51.0i)T2 1 + (-1.97 - 1.14i)T + (29.5 + 51.0i)T^{2}
61 1+(0.743−0.429i)T+(30.5−52.8i)T2 1 + (0.743 - 0.429i)T + (30.5 - 52.8i)T^{2}
67 1+(−9.48+5.47i)T+(33.5−58.0i)T2 1 + (-9.48 + 5.47i)T + (33.5 - 58.0i)T^{2}
71 1+(4.71+8.17i)T+(−35.5+61.4i)T2 1 + (4.71 + 8.17i)T + (-35.5 + 61.4i)T^{2}
73 1+10.2iT−73T2 1 + 10.2iT - 73T^{2}
79 1+(5.04−2.91i)T+(39.5−68.4i)T2 1 + (5.04 - 2.91i)T + (39.5 - 68.4i)T^{2}
83 1+(0.0647+0.0373i)T+(41.5+71.8i)T2 1 + (0.0647 + 0.0373i)T + (41.5 + 71.8i)T^{2}
89 1+(−12.1−6.99i)T+(44.5+77.0i)T2 1 + (-12.1 - 6.99i)T + (44.5 + 77.0i)T^{2}
97 1−2.26T+97T2 1 - 2.26T + 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

−11.04511310295028154055783448654, −10.40803378034916027856048621987, −9.181824152245541333931057140958, −8.566537974940951837622286941019, −7.61799774677143613672069094420, −6.27837572322901815651390078617, −5.15684423151654868919069025401, −3.95353597022706970669004707334, −2.39616603133974675002026766092, −1.72644212642618854903264326448, 2.53976605369694007338187521746, 3.27502055199344352323306096304, 4.90002759560546124507440838448, 5.63335548702366680044480275006, 6.87744868719631774921045313026, 8.193621011692395215483279066023, 8.517928955799831368128188486144, 9.823563956284727607524546293077, 10.37512521450204654439994929029, 11.53540388076746514614202506472

Graph of the ZZ-function along the critical line