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Properties

Label 2-546-7.2-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.832 + 0.553i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.82 − 3.15i)5-s − 0.999·6-s + (1.32 − 2.29i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.82 − 3.15i)10-s + (−0.322 − 0.559i)11-s + (−0.499 + 0.866i)12-s + 13-s + (−1.32 − 2.29i)14-s − 3.64·15-s + (−0.5 + 0.866i)16-s + (3.32 + 5.75i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.815 − 1.41i)5-s − 0.408·6-s + (0.499 − 0.866i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.576 − 0.998i)10-s + (−0.0973 − 0.168i)11-s + (−0.144 + 0.249i)12-s + 0.277·13-s + (−0.353 − 0.612i)14-s − 0.941·15-s + (−0.125 + 0.216i)16-s + (0.805 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518473 - 1.71558i\)
\(L(\frac12)\) \(\approx\) \(0.518473 - 1.71558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.322 + 0.559i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.32 - 5.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.17 + 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.29T + 29T^{2} \)
31 \( 1 + (1.64 + 2.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.82 - 4.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.35T + 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.96 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (8.46 - 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.937T + 97T^{2} \)
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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

−10.41674077742702777093540672677, −9.863085159512108622278239191595, −8.511702225976907984176784010561, −8.105392606019539465949783426465, −6.50616402788532292830954260484, −5.63961522640597014932236067061, −4.79737344508857222874342868426, −3.77448760426233944863762148655, −1.85181221802496771421438043821, −1.06478190262896994897702337836, 2.40681078587600065588831135773, 3.34661642467195392339589268862, 4.94865428543121604725714759219, 5.56694741709042270495821998700, 6.58330317423725566644840118423, 7.23395046971510943436281205721, 8.540489629768682431590279756971, 9.453101685790299503071797231038, 10.24803557040681225666392818332, 11.20114132548503471797263558283

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