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Properties

Label 2-483-161.160-c1-0-3
Degree 22
Conductor 483483
Sign 0.0829−0.996i0.0829 - 0.996i
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  − 2.11·2-s − i·3-s + 2.47·4-s + 1.98·5-s + 2.11i·6-s + (−2.62 + 0.302i)7-s − 1.00·8-s − 9-s − 4.20·10-s + 2.93i·11-s − 2.47i·12-s + 4.75i·13-s + (5.55 − 0.638i)14-s − 1.98i·15-s − 2.83·16-s − 0.444·17-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577i·3-s + 1.23·4-s + 0.889·5-s + 0.863i·6-s + (−0.993 + 0.114i)7-s − 0.353·8-s − 0.333·9-s − 1.33·10-s + 0.883i·11-s − 0.713i·12-s + 1.31i·13-s + (1.48 − 0.170i)14-s − 0.513i·15-s − 0.707·16-s − 0.107·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.0829−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.0829−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.0829−0.996i0.0829 - 0.996i
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), 0.0829−0.996i)(2,\ 483,\ (\ :1/2),\ 0.0829 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.324474+0.298588i0.324474 + 0.298588i
L(12)L(\frac12) ≈\approx 0.324474+0.298588i0.324474 + 0.298588i
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+(2.62−0.302i)T 1 + (2.62 - 0.302i)T
23 1+(−4.70−0.940i)T 1 + (-4.70 - 0.940i)T
good2 1+2.11T+2T2 1 + 2.11T + 2T^{2}
5 1−1.98T+5T2 1 - 1.98T + 5T^{2}
11 1−2.93iT−11T2 1 - 2.93iT - 11T^{2}
13 1−4.75iT−13T2 1 - 4.75iT - 13T^{2}
17 1+0.444T+17T2 1 + 0.444T + 17T^{2}
19 1+6.30T+19T2 1 + 6.30T + 19T^{2}
29 1+2.58T+29T2 1 + 2.58T + 29T^{2}
31 1+6.35iT−31T2 1 + 6.35iT - 31T^{2}
37 1−6.69iT−37T2 1 - 6.69iT - 37T^{2}
41 1−10.5iT−41T2 1 - 10.5iT - 41T^{2}
43 1−12.6iT−43T2 1 - 12.6iT - 43T^{2}
47 1+5.66iT−47T2 1 + 5.66iT - 47T^{2}
53 1−10.6iT−53T2 1 - 10.6iT - 53T^{2}
59 1+6.39iT−59T2 1 + 6.39iT - 59T^{2}
61 1+3.42T+61T2 1 + 3.42T + 61T^{2}
67 1−9.73iT−67T2 1 - 9.73iT - 67T^{2}
71 1−8.75T+71T2 1 - 8.75T + 71T^{2}
73 1+0.926iT−73T2 1 + 0.926iT - 73T^{2}
79 1+8.86iT−79T2 1 + 8.86iT - 79T^{2}
83 1+4.81T+83T2 1 + 4.81T + 83T^{2}
89 1−0.0511T+89T2 1 - 0.0511T + 89T^{2}
97 1−17.4T+97T2 1 - 17.4T + 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

−10.96046009629797589959304157644, −9.844443208512583511135068054958, −9.518159155862584492500478538744, −8.738699298025780362853036623212, −7.63962196380475713871165601832, −6.66415222482631463470044869277, −6.25210836025236319684533930391, −4.49297340888333469216494495704, −2.50684068057011214115807020072, −1.58775319213468803849386458630, 0.42466584081625905436003491518, 2.32578866399791171734492496739, 3.60815073457334114195230687116, 5.36230378401750704956893652798, 6.26668225224520818725246693440, 7.26759034583948714663011294213, 8.552681798607522324838115939812, 8.979205232367181362298375329183, 9.866020536664119414924998023416, 10.61852756982900967923616585466

Graph of the ZZ-function along the critical line