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Properties

Label 2-546-39.2-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.872 - 0.489i$
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 + (−1.60 − 0.640i)3-s + (−0.866 − 0.499i)4-s + (−2.11 − 2.11i)5-s + (1.03 − 1.38i)6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (2.17 + 2.06i)9-s + (2.59 − 1.49i)10-s + (−0.775 − 0.207i)11-s + (1.07 + 1.35i)12-s + (0.382 − 3.58i)13-s + i·14-s + (2.05 + 4.76i)15-s + (0.500 + 0.866i)16-s + (−3.86 + 6.69i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.929 − 0.369i)3-s + (−0.433 − 0.249i)4-s + (−0.947 − 0.947i)5-s + (0.422 − 0.566i)6-s + (0.365 − 0.0978i)7-s + (0.249 − 0.249i)8-s + (0.726 + 0.687i)9-s + (0.820 − 0.473i)10-s + (−0.233 − 0.0626i)11-s + (0.309 + 0.392i)12-s + (0.106 − 0.994i)13-s + 0.267i·14-s + (0.529 + 1.23i)15-s + (0.125 + 0.216i)16-s + (−0.937 + 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.872 - 0.489i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.872 - 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0425094 + 0.162687i\)
\(L(\frac12)\) \(\approx\) \(0.0425094 + 0.162687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 0.640i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 + (-0.382 + 3.58i)T \)
good5 \( 1 + (2.11 + 2.11i)T + 5iT^{2} \)
11 \( 1 + (0.775 + 0.207i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.86 - 6.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.311 + 1.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.763 - 1.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.26 - 3.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.29 - 6.29i)T - 31iT^{2} \)
37 \( 1 + (2.24 - 8.37i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.512 + 1.91i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.83 - 3.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.633 + 0.633i)T - 47iT^{2} \)
53 \( 1 - 6.60iT - 53T^{2} \)
59 \( 1 + (2.31 + 8.65i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.00670 + 0.0116i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 + 2.78i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-13.8 + 3.71i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.65 - 5.65i)T + 73iT^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + (3.82 + 3.82i)T + 83iT^{2} \)
89 \( 1 + (6.45 + 1.72i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.08 - 11.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.97155308209428947988431845677, −10.61929643743899507907989956812, −9.129706587318330850261226458412, −8.235358055490029401379359017311, −7.69681680911543663817070364236, −6.66948124729656643601867421725, −5.58334138562647024032208804736, −4.87127456527083257134540375711, −3.87721214918666490211612263802, −1.39326461840985395407029297060, 0.12381680878037103446431139457, 2.28611903090656585581161076291, 3.77160109117272552407480967924, 4.45936889244674523178060321729, 5.67478030307685593734777116983, 7.01345699341635057216904782960, 7.50309963718397295251115345664, 8.990164057496783562844146036073, 9.678176541422591442551806227685, 10.90329770556643257460920792050

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