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Properties

Label 2-370-185.104-c1-0-5
Degree 22
Conductor 370370
Sign −0.0114−0.999i-0.0114 - 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  + (0.939 − 0.342i)2-s + (−0.400 + 1.10i)3-s + (0.766 − 0.642i)4-s + (−0.0612 + 2.23i)5-s + 1.17i·6-s + (−3.81 + 0.672i)7-s + (0.500 − 0.866i)8-s + (1.24 + 1.04i)9-s + (0.706 + 2.12i)10-s + (−1.57 + 2.73i)11-s + (0.400 + 1.10i)12-s + (−1.40 + 1.17i)13-s + (−3.35 + 1.93i)14-s + (−2.43 − 0.962i)15-s + (0.173 − 0.984i)16-s + (4.46 + 3.74i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (−0.231 + 0.635i)3-s + (0.383 − 0.321i)4-s + (−0.0273 + 0.999i)5-s + 0.477i·6-s + (−1.44 + 0.254i)7-s + (0.176 − 0.306i)8-s + (0.416 + 0.349i)9-s + (0.223 + 0.670i)10-s + (−0.476 + 0.825i)11-s + (0.115 + 0.317i)12-s + (−0.389 + 0.326i)13-s + (−0.896 + 0.517i)14-s + (−0.628 − 0.248i)15-s + (0.0434 − 0.246i)16-s + (1.08 + 0.908i)17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((−0.0114−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0114 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((−0.0114−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0114 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: −0.0114−0.999i-0.0114 - 0.999i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(289,⋅)\chi_{370} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), −0.0114−0.999i)(2,\ 370,\ (\ :1/2),\ -0.0114 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.05382+1.06597i1.05382 + 1.06597i
L(12)L(\frac12) ≈\approx 1.05382+1.06597i1.05382 + 1.06597i
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+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
5 1+(0.0612−2.23i)T 1 + (0.0612 - 2.23i)T
37 1+(−5.80−1.80i)T 1 + (-5.80 - 1.80i)T
good3 1+(0.400−1.10i)T+(−2.29−1.92i)T2 1 + (0.400 - 1.10i)T + (-2.29 - 1.92i)T^{2}
7 1+(3.81−0.672i)T+(6.57−2.39i)T2 1 + (3.81 - 0.672i)T + (6.57 - 2.39i)T^{2}
11 1+(1.57−2.73i)T+(−5.5−9.52i)T2 1 + (1.57 - 2.73i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.40−1.17i)T+(2.25−12.8i)T2 1 + (1.40 - 1.17i)T + (2.25 - 12.8i)T^{2}
17 1+(−4.46−3.74i)T+(2.95+16.7i)T2 1 + (-4.46 - 3.74i)T + (2.95 + 16.7i)T^{2}
19 1+(−2.67+7.36i)T+(−14.5−12.2i)T2 1 + (-2.67 + 7.36i)T + (-14.5 - 12.2i)T^{2}
23 1+(−0.0262−0.0453i)T+(−11.5+19.9i)T2 1 + (-0.0262 - 0.0453i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.42−1.97i)T+(14.5+25.1i)T2 1 + (-3.42 - 1.97i)T + (14.5 + 25.1i)T^{2}
31 1−1.88iT−31T2 1 - 1.88iT - 31T^{2}
41 1+(−2.75+2.30i)T+(7.11−40.3i)T2 1 + (-2.75 + 2.30i)T + (7.11 - 40.3i)T^{2}
43 1−2.80T+43T2 1 - 2.80T + 43T^{2}
47 1+(−9.42+5.44i)T+(23.5−40.7i)T2 1 + (-9.42 + 5.44i)T + (23.5 - 40.7i)T^{2}
53 1+(12.8+2.25i)T+(49.8+18.1i)T2 1 + (12.8 + 2.25i)T + (49.8 + 18.1i)T^{2}
59 1+(−12.1−2.13i)T+(55.4+20.1i)T2 1 + (-12.1 - 2.13i)T + (55.4 + 20.1i)T^{2}
61 1+(−0.0728−0.0867i)T+(−10.5+60.0i)T2 1 + (-0.0728 - 0.0867i)T + (-10.5 + 60.0i)T^{2}
67 1+(−4.55+0.803i)T+(62.9−22.9i)T2 1 + (-4.55 + 0.803i)T + (62.9 - 22.9i)T^{2}
71 1+(12.4+4.52i)T+(54.3+45.6i)T2 1 + (12.4 + 4.52i)T + (54.3 + 45.6i)T^{2}
73 1+4.68iT−73T2 1 + 4.68iT - 73T^{2}
79 1+(5.72−1.01i)T+(74.2−27.0i)T2 1 + (5.72 - 1.01i)T + (74.2 - 27.0i)T^{2}
83 1+(−0.100+0.119i)T+(−14.4−81.7i)T2 1 + (-0.100 + 0.119i)T + (-14.4 - 81.7i)T^{2}
89 1+(−2.80−0.493i)T+(83.6+30.4i)T2 1 + (-2.80 - 0.493i)T + (83.6 + 30.4i)T^{2}
97 1+(8.58+14.8i)T+(−48.5+84.0i)T2 1 + (8.58 + 14.8i)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

−11.61165074981510916074955034359, −10.55895791606754626560184173737, −10.06275504572508759933543610857, −9.358651578408508432118154178426, −7.47823217403235682275702168111, −6.79409199851062583561527158792, −5.73515336435690449647362545474, −4.61340779494552158850755080465, −3.44782085787453599639932572201, −2.49627356114473406586285284795, 0.864788193645494325088382525638, 3.00681161568215514543829956084, 4.09285637138162362504077117580, 5.59720265937827018291246671135, 6.09174734227405604921479696258, 7.36121467980058653793624977183, 8.020208487223055701108663922343, 9.497864621987214171185664444223, 10.06338008452345607328260310091, 11.61473490791800706957817914605

Graph of the ZZ-function along the critical line