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Properties

Label 2-483-161.160-c1-0-26
Degree 22
Conductor 483483
Sign 0.974−0.222i0.974 - 0.222i
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  + 1.86·2-s + i·3-s + 1.46·4-s + 4.10·5-s + 1.86i·6-s + (0.663 − 2.56i)7-s − 8-s − 9-s + 7.63·10-s − 1.89i·11-s + 1.46i·12-s + 2.18i·13-s + (1.23 − 4.76i)14-s + 4.10i·15-s − 4.78·16-s − 1.18·17-s + ⋯
L(s)  = 1  + 1.31·2-s + 0.577i·3-s + 0.731·4-s + 1.83·5-s + 0.759i·6-s + (0.250 − 0.968i)7-s − 0.353·8-s − 0.333·9-s + 2.41·10-s − 0.572i·11-s + 0.422i·12-s + 0.605i·13-s + (0.329 − 1.27i)14-s + 1.05i·15-s − 1.19·16-s − 0.287·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.974−0.222i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.974−0.222i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.974−0.222i0.974 - 0.222i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(160,⋅)\chi_{483} (160, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.974−0.222i)(2,\ 483,\ (\ :1/2),\ 0.974 - 0.222i)

Particular Values

L(1)L(1) ≈\approx 3.23700+0.364448i3.23700 + 0.364448i
L(12)L(\frac12) ≈\approx 3.23700+0.364448i3.23700 + 0.364448i
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−iT 1 - iT
7 1+(−0.663+2.56i)T 1 + (-0.663 + 2.56i)T
23 1+(4.25−2.20i)T 1 + (4.25 - 2.20i)T
good2 1−1.86T+2T2 1 - 1.86T + 2T^{2}
5 1−4.10T+5T2 1 - 4.10T + 5T^{2}
11 1+1.89iT−11T2 1 + 1.89iT - 11T^{2}
13 1−2.18iT−13T2 1 - 2.18iT - 13T^{2}
17 1+1.18T+17T2 1 + 1.18T + 17T^{2}
19 1+4.98T+19T2 1 + 4.98T + 19T^{2}
29 1−2.39T+29T2 1 - 2.39T + 29T^{2}
31 1−9.32iT−31T2 1 - 9.32iT - 31T^{2}
37 1−6.92iT−37T2 1 - 6.92iT - 37T^{2}
41 1+5.60iT−41T2 1 + 5.60iT - 41T^{2}
43 1+5.38iT−43T2 1 + 5.38iT - 43T^{2}
47 1−9.57iT−47T2 1 - 9.57iT - 47T^{2}
53 1+5.85iT−53T2 1 + 5.85iT - 53T^{2}
59 1+3.50iT−59T2 1 + 3.50iT - 59T^{2}
61 1−1.49T+61T2 1 - 1.49T + 61T^{2}
67 1+3.48iT−67T2 1 + 3.48iT - 67T^{2}
71 1−1.81T+71T2 1 - 1.81T + 71T^{2}
73 1+7.97iT−73T2 1 + 7.97iT - 73T^{2}
79 1+14.0iT−79T2 1 + 14.0iT - 79T^{2}
83 1−2.51T+83T2 1 - 2.51T + 83T^{2}
89 1+4.57T+89T2 1 + 4.57T + 89T^{2}
97 1−7.44T+97T2 1 - 7.44T + 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

−10.89396210334417705045263419121, −10.31956295615472589456454446915, −9.371893379835241179300137781598, −8.544366709265978350519451436255, −6.72708297643269860162839058426, −6.16387953942720101013813821129, −5.18043642646086444763224304400, −4.42785553001265736486222604778, −3.27722170801616874437916166542, −1.93490868601404943841463345237, 2.09500060673471962157198010144, 2.59320247479866846798542933645, 4.41489817409829726654798048391, 5.51715871953315008735226006825, 5.97045655189728810642828726224, 6.70632483814990223201872582573, 8.298704454535582249499546184295, 9.216186999861685414984280093226, 10.05700491710957180195557424778, 11.20902706848890629835296441953

Graph of the ZZ-function along the critical line