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Properties

Label 2-546-13.10-c1-0-6
Degree $2$
Conductor $546$
Sign $0.824 + 0.565i$
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 − 0.732i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (1.5 − 0.866i)11-s − 0.999·12-s + (1.59 + 3.23i)13-s − 0.999·14-s + (−0.633 + 0.366i)15-s + (−0.5 − 0.866i)16-s + (1.86 − 3.23i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s − 0.327i·5-s + (0.353 + 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (0.452 − 0.261i)11-s − 0.288·12-s + (0.443 + 0.896i)13-s − 0.267·14-s + (−0.163 + 0.0945i)15-s + (−0.125 − 0.216i)16-s + (0.452 − 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.824 + 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990612 - 0.306880i\)
\(L(\frac12)\) \(\approx\) \(0.990612 - 0.306880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 3.23i)T \)
good5 \( 1 + 0.732iT - 5T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \)
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 \( 1 + (3.23 + 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.19iT - 31T^{2} \)
37 \( 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.63 + 2.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.46iT - 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + (-0.803 - 0.464i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.59 - 4.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.73 + 4.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.90 + 1.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.46iT - 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 + 0.196iT - 83T^{2} \)
89 \( 1 + (9.06 - 5.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.79085251521530519987053866857, −9.607515564867724926863919977306, −8.910811064175830418972629874358, −8.049184739844742437751944333469, −7.15873600195455352732704127888, −6.25984439166830131523684117458, −5.38567503252472026963338892338, −4.14794305922382447816013905362, −2.32096330944653758915800400891, −0.926301273700834640319346484287, 1.29360346199130214710424497254, 3.02685845403989968944866945151, 4.00716912079894669915291008645, 5.30416635511018551316427233145, 6.37243157379974201728589590801, 7.45635512146812402965429340057, 8.328420958027490393525284725526, 9.265719529422004260797475720291, 10.14101182143248310448590899383, 10.79138033454868803260646687652

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