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Properties

Label 2-182-13.12-c1-0-2
Degree $2$
Conductor $182$
Sign $-0.0233 - 0.999i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + i·2-s + 1.52·3-s − 4-s + 4.16i·5-s + 1.52i·6-s − i·7-s − i·8-s − 0.688·9-s − 4.16·10-s − 4.64i·11-s − 1.52·12-s + (0.0843 + 3.60i)13-s + 14-s + 6.33i·15-s + 16-s + 6.33·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.877·3-s − 0.5·4-s + 1.86i·5-s + 0.620i·6-s − 0.377i·7-s − 0.353i·8-s − 0.229·9-s − 1.31·10-s − 1.40i·11-s − 0.438·12-s + (0.0233 + 0.999i)13-s + 0.267·14-s + 1.63i·15-s + 0.250·16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0233 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0233 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.0233 - 0.999i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ -0.0233 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975910 + 0.999010i\)
\(L(\frac12)\) \(\approx\) \(0.975910 + 0.999010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + iT \)
13 \( 1 + (-0.0843 - 3.60i)T \)
good3 \( 1 - 1.52T + 3T^{2} \)
5 \( 1 - 4.16iT - 5T^{2} \)
11 \( 1 + 4.64iT - 11T^{2} \)
17 \( 1 - 6.33T + 17T^{2} \)
19 \( 1 + 2.16iT - 19T^{2} \)
23 \( 1 - 3.68T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 - 1.68iT - 37T^{2} \)
41 \( 1 + 9.68iT - 41T^{2} \)
43 \( 1 + 4.33T + 43T^{2} \)
47 \( 1 + 3.35iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 9.54iT - 59T^{2} \)
61 \( 1 - 0.479T + 61T^{2} \)
67 \( 1 - 0.311iT - 67T^{2} \)
71 \( 1 + 6.08iT - 71T^{2} \)
73 \( 1 - 1.35iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 11.4iT - 83T^{2} \)
89 \( 1 - 1.66iT - 89T^{2} \)
97 \( 1 + 2.31iT - 97T^{2} \)
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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

−13.62405677943352228390227679644, −11.70420852337368912725186954952, −10.84743716981368254866687861337, −9.807201275820651700103343000062, −8.659304720059987047037101468267, −7.66141112160342595398190756070, −6.79232533876275435002204542431, −5.78873243066428453348730183154, −3.68606491323317005485094810225, −2.86526960208275699524225764029, 1.47175011651461004066187776997, 3.13493755301173827709043128841, 4.65569962924037068855490432691, 5.51621091837391478117587636996, 7.85791140172067571607550815689, 8.430749024906641646623153031276, 9.420666715777001667592578038828, 10.04362747285265651607986858340, 11.77598934474117168128917714919, 12.58621480881133477929492751646

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