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Properties

Label 2-370-185.23-c1-0-4
Degree 22
Conductor 370370
Sign −0.0126−0.999i-0.0126 - 0.999i
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 + (−2.83 − 0.758i)3-s + (0.499 + 0.866i)4-s + (−1.91 − 1.16i)5-s + (−2.07 − 2.07i)6-s + (2.40 + 0.644i)7-s + 0.999i·8-s + (4.84 + 2.79i)9-s + (−1.07 − 1.96i)10-s + 2.91i·11-s + (−0.758 − 2.83i)12-s + (−2.54 + 1.47i)13-s + (1.76 + 1.76i)14-s + (4.53 + 4.73i)15-s + (−0.5 + 0.866i)16-s + (−2.25 + 3.89i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.63 − 0.438i)3-s + (0.249 + 0.433i)4-s + (−0.854 − 0.518i)5-s + (−0.846 − 0.846i)6-s + (0.909 + 0.243i)7-s + 0.353i·8-s + (1.61 + 0.932i)9-s + (−0.340 − 0.619i)10-s + 0.878i·11-s + (−0.219 − 0.817i)12-s + (−0.706 + 0.407i)13-s + (0.470 + 0.470i)14-s + (1.17 + 1.22i)15-s + (−0.125 + 0.216i)16-s + (−0.545 + 0.945i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.0126−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.0126−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.0126−0.999i-0.0126 - 0.999i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(23,⋅)\chi_{370} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.0126−0.999i)(2,\ 370,\ (\ :1/2),\ -0.0126 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.591331+0.598884i0.591331 + 0.598884i
L(12)L(\frac12) ≈\approx 0.591331+0.598884i0.591331 + 0.598884i
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+(1.91+1.16i)T 1 + (1.91 + 1.16i)T
37 1+(5.53+2.51i)T 1 + (5.53 + 2.51i)T
good3 1+(2.83+0.758i)T+(2.59+1.5i)T2 1 + (2.83 + 0.758i)T + (2.59 + 1.5i)T^{2}
7 1+(−2.40−0.644i)T+(6.06+3.5i)T2 1 + (-2.40 - 0.644i)T + (6.06 + 3.5i)T^{2}
11 1−2.91iT−11T2 1 - 2.91iT - 11T^{2}
13 1+(2.54−1.47i)T+(6.5−11.2i)T2 1 + (2.54 - 1.47i)T + (6.5 - 11.2i)T^{2}
17 1+(2.25−3.89i)T+(−8.5−14.7i)T2 1 + (2.25 - 3.89i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.54−0.949i)T+(16.4+9.5i)T2 1 + (-3.54 - 0.949i)T + (16.4 + 9.5i)T^{2}
23 1−6.99iT−23T2 1 - 6.99iT - 23T^{2}
29 1+(−3.87−3.87i)T+29iT2 1 + (-3.87 - 3.87i)T + 29iT^{2}
31 1+(−3.48+3.48i)T−31iT2 1 + (-3.48 + 3.48i)T - 31iT^{2}
41 1+(4.86−2.81i)T+(20.5−35.5i)T2 1 + (4.86 - 2.81i)T + (20.5 - 35.5i)T^{2}
43 1+9.87iT−43T2 1 + 9.87iT - 43T^{2}
47 1+(−6.90+6.90i)T−47iT2 1 + (-6.90 + 6.90i)T - 47iT^{2}
53 1+(9.56−2.56i)T+(45.8−26.5i)T2 1 + (9.56 - 2.56i)T + (45.8 - 26.5i)T^{2}
59 1+(−1.43−5.34i)T+(−51.0+29.5i)T2 1 + (-1.43 - 5.34i)T + (-51.0 + 29.5i)T^{2}
61 1+(−4.95−1.32i)T+(52.8+30.5i)T2 1 + (-4.95 - 1.32i)T + (52.8 + 30.5i)T^{2}
67 1+(0.901−3.36i)T+(−58.0−33.5i)T2 1 + (0.901 - 3.36i)T + (-58.0 - 33.5i)T^{2}
71 1+(−8.12−14.0i)T+(−35.5+61.4i)T2 1 + (-8.12 - 14.0i)T + (-35.5 + 61.4i)T^{2}
73 1+(1.91−1.91i)T−73iT2 1 + (1.91 - 1.91i)T - 73iT^{2}
79 1+(4.93+1.32i)T+(68.4+39.5i)T2 1 + (4.93 + 1.32i)T + (68.4 + 39.5i)T^{2}
83 1+(9.27−2.48i)T+(71.8−41.5i)T2 1 + (9.27 - 2.48i)T + (71.8 - 41.5i)T^{2}
89 1+(5.84−1.56i)T+(77.0−44.5i)T2 1 + (5.84 - 1.56i)T + (77.0 - 44.5i)T^{2}
97 1+1.22T+97T2 1 + 1.22T + 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

−11.79761906867292394290614795305, −11.24196773385791024332824363126, −10.09539262554667786643442131979, −8.557320748721915834299138342092, −7.44697383761389853763482593287, −6.92741399300898700101471194724, −5.53519750641329171507309745761, −4.98847505248608064476290691701, −4.08729300353729802799024719379, −1.63588329524304691184460085878, 0.60158473974736866851833754022, 3.04193732655950193468268250663, 4.59661617493827347589151797128, 4.89038060713901952023247000636, 6.19930440876893380435223389813, 7.03637612362686480728213129143, 8.214450194582396736602525545782, 9.861342467095446045927449749767, 10.78808367462859863215618302286, 11.17464377053694444467113680233

Graph of the ZZ-function along the critical line