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Properties

Label 2-546-13.9-c1-0-4
Degree $2$
Conductor $546$
Sign $0.859 - 0.511i$
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 − 3.27·5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.63 − 2.83i)10-s + (2 − 3.46i)11-s + 0.999·12-s + (3.5 + 0.866i)13-s + 0.999·14-s + (1.63 − 2.83i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.46·5-s + (−0.204 − 0.353i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.517 − 0.896i)10-s + (0.603 − 1.04i)11-s + 0.288·12-s + (0.970 + 0.240i)13-s + 0.267·14-s + (0.422 − 0.732i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761390 + 0.209255i\)
\(L(\frac12)\) \(\approx\) \(0.761390 + 0.209255i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.27 - 7.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.13 - 1.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.362 - 0.627i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.27T + 31T^{2} \)
37 \( 1 + (-3.63 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.362 - 0.627i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.41 + 9.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.54T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + (5.41 + 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.13 + 8.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.13 - 1.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (4.13 - 7.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.27 + 12.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93984887230098916820572081510, −9.957039420396949359889835836421, −8.919060818049423603908099878907, −8.214573436989799727686142794745, −7.38474402246854313751929396183, −6.36180696536628095206601972150, −5.42282855183347354190328533060, −3.99861042369625814833989955675, −3.58636921122831018395502785470, −0.77980177529961547556015709261, 0.972198847389114395569722339515, 2.68899801264906544670085670522, 3.91698530874514036110549935709, 4.79920863205867065813416041662, 6.42328422609142078290890984412, 7.23973489423363799615641590840, 8.144325962756864746498641701013, 8.848196750661737564426410476772, 9.916071634235912576587484597276, 11.03712851761541453478982750309

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