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Properties

Label 2-38e2-76.43-c0-0-2
Degree 22
Conductor 14441444
Sign 0.872+0.489i0.872 + 0.489i
Analytic cond. 0.7206490.720649
Root an. cond. 0.8489100.848910
Motivic weight 00
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 + (0.766 − 0.642i)4-s + (0.473 + 0.397i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.580 + 0.211i)10-s + (0.280 + 1.59i)13-s + (0.173 − 0.984i)16-s + (1.52 − 0.553i)17-s − 18-s + 0.618·20-s + (−0.107 − 0.608i)25-s + (0.809 + 1.40i)26-s + (−1.52 − 0.553i)29-s + (−0.173 − 0.984i)32-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.473 + 0.397i)5-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.580 + 0.211i)10-s + (0.280 + 1.59i)13-s + (0.173 − 0.984i)16-s + (1.52 − 0.553i)17-s − 18-s + 0.618·20-s + (−0.107 − 0.608i)25-s + (0.809 + 1.40i)26-s + (−1.52 − 0.553i)29-s + (−0.173 − 0.984i)32-s + ⋯

Functional equation

Λ(s)=(1444s/2 ΓC(s) L(s)=((0.872+0.489i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1444s/2 ΓC(s) L(s)=((0.872+0.489i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14441444    =    22⋅1922^{2} \cdot 19^{2}
Sign: 0.872+0.489i0.872 + 0.489i
Analytic conductor: 0.7206490.720649
Root analytic conductor: 0.8489100.848910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1444(423,⋅)\chi_{1444} (423, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1444, ( :0), 0.872+0.489i)(2,\ 1444,\ (\ :0),\ 0.872 + 0.489i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0079509492.007950949
L(12)L(\frac12) ≈\approx 2.0079509492.007950949
L(1)L(1) 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
19 1 1
good3 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
5 1+(−0.473−0.397i)T+(0.173+0.984i)T2 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(−0.280−1.59i)T+(−0.939+0.342i)T2 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2}
17 1+(−1.52+0.553i)T+(0.766−0.642i)T2 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2}
23 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
29 1+(1.52+0.553i)T+(0.766+0.642i)T2 1 + (1.52 + 0.553i)T + (0.766 + 0.642i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+0.618T+T2 1 + 0.618T + T^{2}
41 1+(0.107−0.608i)T+(−0.939−0.342i)T2 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2}
43 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
47 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
53 1+(0.473−0.397i)T+(0.173−0.984i)T2 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(1.23−1.04i)T+(0.173−0.984i)T2 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2}
67 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
71 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
73 1+(−0.107+0.608i)T+(−0.939−0.342i)T2 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2}
79 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(0.107+0.608i)T+(−0.939+0.342i)T2 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2}
97 1+(1.52−0.553i)T+(0.766−0.642i)T2 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)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

−9.671332521210169977892011127355, −9.174601749312054818087718780487, −7.88436225515720016165497696312, −6.96312926370642743081648677156, −6.13939079854093484665350719257, −5.63056150096935075320905505067, −4.54347760472291183901769002211, −3.55805458109715855691361100858, −2.71126620787086906439744874980, −1.61205313937249591726604730771, 1.72375609122142383941560053139, 3.07600560339644968896983209855, 3.64117185564408647250120226607, 5.28570000664308299139072990897, 5.38399411451533275030919395810, 6.17128619299666342245971746841, 7.43182598798615811657779144185, 8.025976441242058752266083692730, 8.742523088544711655132981286625, 9.878791088042973451981796150152

Graph of the ZZ-function along the critical line