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Properties

Label 2-273-13.12-c1-0-7
Degree 22
Conductor 273273
Sign 0.996+0.0862i0.996 + 0.0862i
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  − 0.311i·2-s + 3-s + 1.90·4-s + 1.52i·5-s − 0.311i·6-s − i·7-s − 1.21i·8-s + 9-s + 0.474·10-s − 1.09i·11-s + 1.90·12-s + (−0.311 + 3.59i)13-s − 0.311·14-s + 1.52i·15-s + 3.42·16-s − 4.42·17-s + ⋯
L(s)  = 1  − 0.219i·2-s + 0.577·3-s + 0.951·4-s + 0.682i·5-s − 0.127i·6-s − 0.377i·7-s − 0.429i·8-s + 0.333·9-s + 0.150·10-s − 0.330i·11-s + 0.549·12-s + (−0.0862 + 0.996i)13-s − 0.0831·14-s + 0.393i·15-s + 0.857·16-s − 1.07·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.996+0.0862i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.996+0.0862i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.996+0.0862i0.996 + 0.0862i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(64,⋅)\chi_{273} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.996+0.0862i)(2,\ 273,\ (\ :1/2),\ 0.996 + 0.0862i)

Particular Values

L(1)L(1) ≈\approx 1.80138−0.0778623i1.80138 - 0.0778623i
L(12)L(\frac12) ≈\approx 1.80138−0.0778623i1.80138 - 0.0778623i
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−T 1 - T
7 1+iT 1 + iT
13 1+(0.311−3.59i)T 1 + (0.311 - 3.59i)T
good2 1+0.311iT−2T2 1 + 0.311iT - 2T^{2}
5 1−1.52iT−5T2 1 - 1.52iT - 5T^{2}
11 1+1.09iT−11T2 1 + 1.09iT - 11T^{2}
17 1+4.42T+17T2 1 + 4.42T + 17T^{2}
19 1+1.80iT−19T2 1 + 1.80iT - 19T^{2}
23 1+3.80T+23T2 1 + 3.80T + 23T^{2}
29 1+0.755T+29T2 1 + 0.755T + 29T^{2}
31 1+4.85iT−31T2 1 + 4.85iT - 31T^{2}
37 1+5.80iT−37T2 1 + 5.80iT - 37T^{2}
41 1−11.3iT−41T2 1 - 11.3iT - 41T^{2}
43 1+5.24T+43T2 1 + 5.24T + 43T^{2}
47 1+2.28iT−47T2 1 + 2.28iT - 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+0.474iT−59T2 1 + 0.474iT - 59T^{2}
61 1+13.0T+61T2 1 + 13.0T + 61T^{2}
67 1−9.80iT−67T2 1 - 9.80iT - 67T^{2}
71 1+13.0iT−71T2 1 + 13.0iT - 71T^{2}
73 1−3.47iT−73T2 1 - 3.47iT - 73T^{2}
79 1+5.37T+79T2 1 + 5.37T + 79T^{2}
83 1−13.8iT−83T2 1 - 13.8iT - 83T^{2}
89 1+13.1iT−89T2 1 + 13.1iT - 89T^{2}
97 1−4.42iT−97T2 1 - 4.42iT - 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

−11.59873026205693901627568150511, −11.06200879097910046834464087576, −10.13017352691567721280690188406, −9.117166389826995029765596737245, −7.86527482384509222937189736622, −6.93292181150739007290105923746, −6.25347801892522008576555988528, −4.34118793921069571939102553210, −3.09317382451000324356072312721, −1.97851998649247369470744504369, 1.85970556634421356119870076599, 3.16320434466550192932125757184, 4.78811076197096348677920127861, 5.95999730114951667267006316850, 7.08540439103700080986526913782, 8.098853156110072846143139536741, 8.824733503730826407723558036240, 10.04249878460990641182770542750, 10.92118748675425819245008277863, 12.13972071819340141174734084033

Graph of the ZZ-function along the critical line