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Properties

Label 2-546-91.20-c1-0-3
Degree $2$
Conductor $546$
Sign $0.495 - 0.868i$
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  + (−0.965 + 0.258i)2-s + (0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.781 + 0.781i)5-s + (−0.965 − 0.258i)6-s + (−1.20 − 2.35i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.552 − 0.957i)10-s + (0.312 + 1.16i)11-s + 12-s + (2.62 + 2.47i)13-s + (1.77 + 1.96i)14-s + (−1.06 + 0.286i)15-s + (0.500 − 0.866i)16-s + (1.19 + 2.06i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.349 + 0.349i)5-s + (−0.394 − 0.105i)6-s + (−0.455 − 0.890i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (0.174 − 0.302i)10-s + (0.0943 + 0.352i)11-s + 0.288·12-s + (0.728 + 0.685i)13-s + (0.474 + 0.524i)14-s + (−0.275 + 0.0739i)15-s + (0.125 − 0.216i)16-s + (0.289 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.495 - 0.868i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.997089 + 0.579286i\)
\(L(\frac12)\) \(\approx\) \(0.997089 + 0.579286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.20 + 2.35i)T \)
13 \( 1 + (-2.62 - 2.47i)T \)
good5 \( 1 + (0.781 - 0.781i)T - 5iT^{2} \)
11 \( 1 + (-0.312 - 1.16i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.19 - 2.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.87 - 1.30i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.04 - 2.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.18 - 7.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.07 + 6.07i)T - 31iT^{2} \)
37 \( 1 + (2.63 + 9.84i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.0706 - 0.263i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.39 - 5.42i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.79 - 4.79i)T + 47iT^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 + (-2.46 + 9.18i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.83 + 5.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.76 - 1.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.24 - 8.36i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.16 - 4.16i)T + 73iT^{2} \)
79 \( 1 - 7.38T + 79T^{2} \)
83 \( 1 + (2.18 - 2.18i)T - 83iT^{2} \)
89 \( 1 + (15.1 - 4.06i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.77 + 0.476i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.91709897156650687690928764696, −9.820307063576630829528956573637, −9.359990675596747639055249085290, −8.279705259838595649748999654668, −7.34406947838835145907149890870, −6.83774006526858761268593234124, −5.47853883598342973479793706198, −3.98892370930322398747015491191, −3.20532049838466860509864606773, −1.42142317384242860888758785788, 0.908092114782060731933739743721, 2.63795719392355749902467091398, 3.45390195856179017977450573509, 5.10897525615272749029011626271, 6.25638924758564074789762533072, 7.22356797520281720297229161107, 8.388536613089310075835724870581, 8.642867730547323211126653400712, 9.661487139851545419032798830281, 10.45987590203301399263388397925

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