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Properties

Label 2-546-13.3-c1-0-3
Degree $2$
Conductor $546$
Sign $0.342 - 0.939i$
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 − 2·5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (2.58 + 4.48i)11-s − 0.999·12-s + (−1.5 − 3.27i)13-s − 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (0.780 + 1.35i)11-s − 0.288·12-s + (−0.416 − 0.909i)13-s − 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734396 + 0.513764i\)
\(L(\frac12)\) \(\approx\) \(0.734396 + 0.513764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (1.5 + 3.27i)T \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + (-2.58 - 4.48i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.58 - 2.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.08 - 7.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 - 2.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.58 - 6.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.08 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-6.08 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.91 - 3.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.08 + 1.88i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 6.17T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.17 + 8.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.97824405833601616701795341155, −10.00230279821300597157895990156, −9.475998439311390433559590264202, −8.240579418026804574474824537766, −7.76659180068789645909828552232, −6.64646963877933473075456209287, −4.94494751860031872216306115013, −4.11893531363589770482950566249, −3.26423483179434402926065797716, −1.66152350960272933531137111645, 0.58423465200281592779347530177, 2.51635319766143103932682499058, 3.97482456923709664462515045886, 5.04419061149606610582208603161, 6.49743424848565371082611113071, 6.90302359436369252990671801839, 8.066906542196925941207420155090, 8.736498086756324509579271754328, 9.278858989398728013863616779979, 10.73919597840634787213072779463

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