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Properties

Label 2-483-69.68-c1-0-30
Degree 22
Conductor 483483
Sign 0.387+0.921i0.387 + 0.921i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
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  − 0.377i·2-s + (−1.03 − 1.38i)3-s + 1.85·4-s + 2.37·5-s + (−0.523 + 0.392i)6-s − i·7-s − 1.45i·8-s + (−0.840 + 2.87i)9-s − 0.895i·10-s + 6.18·11-s + (−1.93 − 2.57i)12-s − 3.83·13-s − 0.377·14-s + (−2.46 − 3.28i)15-s + 3.16·16-s + 0.877·17-s + ⋯
L(s)  = 1  − 0.267i·2-s + (−0.599 − 0.800i)3-s + 0.928·4-s + 1.06·5-s + (−0.213 + 0.160i)6-s − 0.377i·7-s − 0.515i·8-s + (−0.280 + 0.959i)9-s − 0.283i·10-s + 1.86·11-s + (−0.557 − 0.742i)12-s − 1.06·13-s − 0.100·14-s + (−0.636 − 0.848i)15-s + 0.791·16-s + 0.212·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.387+0.921i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.387+0.921i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.387+0.921i0.387 + 0.921i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(344,⋅)\chi_{483} (344, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.387+0.921i)(2,\ 483,\ (\ :1/2),\ 0.387 + 0.921i)

Particular Values

L(1)L(1) ≈\approx 1.48032−0.983439i1.48032 - 0.983439i
L(12)L(\frac12) ≈\approx 1.48032−0.983439i1.48032 - 0.983439i
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)
bad3 1+(1.03+1.38i)T 1 + (1.03 + 1.38i)T
7 1+iT 1 + iT
23 1+(1.16−4.65i)T 1 + (1.16 - 4.65i)T
good2 1+0.377iT−2T2 1 + 0.377iT - 2T^{2}
5 1−2.37T+5T2 1 - 2.37T + 5T^{2}
11 1−6.18T+11T2 1 - 6.18T + 11T^{2}
13 1+3.83T+13T2 1 + 3.83T + 13T^{2}
17 1−0.877T+17T2 1 - 0.877T + 17T^{2}
19 1−3.29iT−19T2 1 - 3.29iT - 19T^{2}
29 1+5.12iT−29T2 1 + 5.12iT - 29T^{2}
31 1+8.93T+31T2 1 + 8.93T + 31T^{2}
37 1+10.9iT−37T2 1 + 10.9iT - 37T^{2}
41 1−1.00iT−41T2 1 - 1.00iT - 41T^{2}
43 1−2.83iT−43T2 1 - 2.83iT - 43T^{2}
47 1−1.64iT−47T2 1 - 1.64iT - 47T^{2}
53 1+4.73T+53T2 1 + 4.73T + 53T^{2}
59 1+10.7iT−59T2 1 + 10.7iT - 59T^{2}
61 1−6.29iT−61T2 1 - 6.29iT - 61T^{2}
67 1−2.23iT−67T2 1 - 2.23iT - 67T^{2}
71 1−8.32iT−71T2 1 - 8.32iT - 71T^{2}
73 1+9.99T+73T2 1 + 9.99T + 73T^{2}
79 1−13.6iT−79T2 1 - 13.6iT - 79T^{2}
83 1−7.43T+83T2 1 - 7.43T + 83T^{2}
89 1+3.93T+89T2 1 + 3.93T + 89T^{2}
97 1+11.1iT−97T2 1 + 11.1iT - 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

−11.02075984150405302035775455493, −9.969108770833249496135995175090, −9.357351942837371964757391932560, −7.72296437080779177341331727309, −7.03754449836347954389798954856, −6.18756690858637431584771559759, −5.55098703501156032804501683700, −3.83819417059256529281350961784, −2.18083772681073560065012617356, −1.40404223328531844167709526115, 1.75179281478078504141132368640, 3.16724118041299511939607925432, 4.66827153599803119335603821182, 5.67393683245469991343924698172, 6.42183725600406711508892611124, 7.08487974602247674511081066478, 8.771281249911480737223528379575, 9.463181167988548191557922943296, 10.23260262671187727456100445408, 11.12910144237658542947432791965

Graph of the ZZ-function along the critical line