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Properties

Label 2-546-91.76-c1-0-7
Degree $2$
Conductor $546$
Sign $0.511 - 0.859i$
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  + (−0.258 + 0.965i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.201 − 0.201i)5-s + (0.258 + 0.965i)6-s + (−0.864 + 2.50i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.142 + 0.247i)10-s + (3.76 + 1.00i)11-s − 12-s + (3.08 − 1.86i)13-s + (−2.19 − 1.48i)14-s + (0.0738 − 0.275i)15-s + (0.500 + 0.866i)16-s + (−1.53 + 2.65i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (0.0902 − 0.0902i)5-s + (0.105 + 0.394i)6-s + (−0.326 + 0.945i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.0451 + 0.0781i)10-s + (1.13 + 0.303i)11-s − 0.288·12-s + (0.856 − 0.516i)13-s + (−0.585 − 0.396i)14-s + (0.0190 − 0.0711i)15-s + (0.125 + 0.216i)16-s + (−0.371 + 0.644i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.511 - 0.859i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36958 + 0.778197i\)
\(L(\frac12)\) \(\approx\) \(1.36958 + 0.778197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.864 - 2.50i)T \)
13 \( 1 + (-3.08 + 1.86i)T \)
good5 \( 1 + (-0.201 + 0.201i)T - 5iT^{2} \)
11 \( 1 + (-3.76 - 1.00i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.53 - 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.152 + 0.568i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.501 - 0.289i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.10 - 8.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.91 + 6.91i)T - 31iT^{2} \)
37 \( 1 + (7.62 + 2.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.90 - 2.11i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.751 - 0.433i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.05 - 1.05i)T + 47iT^{2} \)
53 \( 1 - 9.83T + 53T^{2} \)
59 \( 1 + (12.2 - 3.27i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.76 - 1.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 12.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.43 - 1.99i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.7 + 10.7i)T + 73iT^{2} \)
79 \( 1 + 9.59T + 79T^{2} \)
83 \( 1 + (10.2 - 10.2i)T - 83iT^{2} \)
89 \( 1 + (-3.58 + 13.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.65 + 9.92i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.84466124279047456649073382977, −9.727392889701630931762061746299, −8.886114918684684343442100441004, −8.519206833258065271242140830409, −7.32066106781011906380335617344, −6.39326024951531451730597345821, −5.71555501952209619471769618997, −4.32676333666231610408723664607, −3.11029996162811919029438854762, −1.50397108928352843524354334117, 1.11875146319623267154388951303, 2.71671769071465885102978145036, 3.88945707200169226850853016697, 4.46012454968070166233165992372, 6.22281654904715153807826456950, 7.05525495137268539093259990595, 8.336824150571282675802261721162, 8.949575052954200524670043902782, 9.934318394615388091570971730201, 10.47532598758868775236782795618

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