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Properties

Label 2-273-13.12-c1-0-10
Degree $2$
Conductor $273$
Sign $0.996 - 0.0862i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.996 - 0.0862i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.996 - 0.0862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80138 + 0.0778623i\)
\(L(\frac12)\) \(\approx\) \(1.80138 + 0.0778623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (0.311 + 3.59i)T \)
good2 \( 1 - 0.311iT - 2T^{2} \)
5 \( 1 + 1.52iT - 5T^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
17 \( 1 + 4.42T + 17T^{2} \)
19 \( 1 - 1.80iT - 19T^{2} \)
23 \( 1 + 3.80T + 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 - 4.85iT - 31T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 2.28iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 0.474iT - 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 9.80iT - 67T^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 + 3.47iT - 73T^{2} \)
79 \( 1 + 5.37T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 4.42iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line