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Properties

Label 2-483-69.68-c1-0-31
Degree 22
Conductor 483483
Sign 0.753+0.656i0.753 + 0.656i
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  − 1.06i·2-s + (−1.10 + 1.33i)3-s + 0.872·4-s + 4.21·5-s + (1.41 + 1.16i)6-s − i·7-s − 3.05i·8-s + (−0.576 − 2.94i)9-s − 4.47i·10-s − 2.41·11-s + (−0.960 + 1.16i)12-s + 4.65·13-s − 1.06·14-s + (−4.63 + 5.63i)15-s − 1.49·16-s − 3.38·17-s + ⋯
L(s)  = 1  − 0.750i·2-s + (−0.635 + 0.772i)3-s + 0.436·4-s + 1.88·5-s + (0.579 + 0.477i)6-s − 0.377i·7-s − 1.07i·8-s + (−0.192 − 0.981i)9-s − 1.41i·10-s − 0.727·11-s + (−0.277 + 0.336i)12-s + 1.29·13-s − 0.283·14-s + (−1.19 + 1.45i)15-s − 0.373·16-s − 0.821·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.753+0.656i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.753+0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.753+0.656i0.753 + 0.656i
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), 0.753+0.656i)(2,\ 483,\ (\ :1/2),\ 0.753 + 0.656i)

Particular Values

L(1)L(1) ≈\approx 1.68179−0.629989i1.68179 - 0.629989i
L(12)L(\frac12) ≈\approx 1.68179−0.629989i1.68179 - 0.629989i
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+(1.10−1.33i)T 1 + (1.10 - 1.33i)T
7 1+iT 1 + iT
23 1+(4.79+0.134i)T 1 + (4.79 + 0.134i)T
good2 1+1.06iT−2T2 1 + 1.06iT - 2T^{2}
5 1−4.21T+5T2 1 - 4.21T + 5T^{2}
11 1+2.41T+11T2 1 + 2.41T + 11T^{2}
13 1−4.65T+13T2 1 - 4.65T + 13T^{2}
17 1+3.38T+17T2 1 + 3.38T + 17T^{2}
19 1−0.938iT−19T2 1 - 0.938iT - 19T^{2}
29 1−2.21iT−29T2 1 - 2.21iT - 29T^{2}
31 1+0.993T+31T2 1 + 0.993T + 31T^{2}
37 1+0.131iT−37T2 1 + 0.131iT - 37T^{2}
41 1−7.16iT−41T2 1 - 7.16iT - 41T^{2}
43 1−4.89iT−43T2 1 - 4.89iT - 43T^{2}
47 1+7.21iT−47T2 1 + 7.21iT - 47T^{2}
53 1−2.35T+53T2 1 - 2.35T + 53T^{2}
59 1+12.6iT−59T2 1 + 12.6iT - 59T^{2}
61 1−4.95iT−61T2 1 - 4.95iT - 61T^{2}
67 1−4.58iT−67T2 1 - 4.58iT - 67T^{2}
71 1−16.5iT−71T2 1 - 16.5iT - 71T^{2}
73 1−3.32T+73T2 1 - 3.32T + 73T^{2}
79 1−8.91iT−79T2 1 - 8.91iT - 79T^{2}
83 1−3.38T+83T2 1 - 3.38T + 83T^{2}
89 1+15.6T+89T2 1 + 15.6T + 89T^{2}
97 1−15.8iT−97T2 1 - 15.8iT - 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

−10.83027705805911878363817044105, −10.08893006413966842560379528611, −9.667294645915899584244651581302, −8.546841549322179813468255262532, −6.74022551926998445545605087640, −6.14754634858023527373327384063, −5.29334682546050017620965206229, −3.93394006691073745919269488084, −2.64944989610444028562922388633, −1.39524820675073010646295446414, 1.75199843619176871290613119114, 2.51485753893589236903997010026, 5.04593103178906288309585021178, 5.97722703276895813602864558894, 6.11902616216118754395225606515, 7.13387011176401153658782828198, 8.255402673126450818616690815485, 9.111355918294679397061349765607, 10.46564420899666550088719327887, 10.89212249476168359646476253154

Graph of the ZZ-function along the critical line