Properties

Label 2-546-13.4-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.839 - 0.543i$
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  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + 2.73i·5-s + (−0.866 + 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.36 + 2.36i)10-s + (1.5 + 0.866i)11-s − 0.999·12-s + (−3.59 − 0.232i)13-s − 0.999·14-s + (−2.36 − 1.36i)15-s + (−0.5 + 0.866i)16-s + (0.133 + 0.232i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 1.22i·5-s + (−0.353 + 0.204i)6-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.431 + 0.748i)10-s + (0.452 + 0.261i)11-s − 0.288·12-s + (−0.997 − 0.0643i)13-s − 0.267·14-s + (−0.610 − 0.352i)15-s + (−0.125 + 0.216i)16-s + (0.0324 + 0.0562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442438 + 1.49578i\)
\(L(\frac12)\) \(\approx\) \(0.442438 + 1.49578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (3.59 + 0.232i)T \)
good5 \( 1 - 2.73iT - 5T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.133 - 0.232i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.232 + 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
37 \( 1 + (2.83 + 1.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.36 - 5.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.46iT - 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + (-11.1 + 6.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.26 - 2.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.09 - 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.46iT - 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (-3.06 - 1.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.43 + 0.830i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11039787865016242311655346931, −10.39673311818028076658445184421, −9.588126184217985442566680849201, −8.445452173229960592104008574833, −7.10126081181171939598290360546, −6.71584660343762529423453692116, −5.62999760687918158943046215073, −4.59782195039248035992870956019, −3.46838720640047242092413157342, −2.51575865112634419861401295750, 0.76537748525508855933771343786, 2.20003898206132195160429842786, 3.75889200576746739129227270549, 4.84680772879350861158005950088, 5.58688539823967146130806279628, 6.68203388395863159604548408452, 7.63007045552729697840034658127, 8.788498513450012242321493952323, 9.575435700663217565883786691136, 10.55511642297210088131039045763

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