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Properties

Label 2-546-91.20-c1-0-12
Degree $2$
Conductor $546$
Sign $0.161 + 0.986i$
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 + (1.77 − 1.77i)5-s + (0.965 + 0.258i)6-s + (1.76 − 1.96i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.25 + 2.17i)10-s + (−0.239 − 0.893i)11-s − 12-s + (1.53 − 3.26i)13-s + (−1.19 + 2.35i)14-s + (−2.42 + 0.649i)15-s + (0.500 − 0.866i)16-s + (2.62 + 4.54i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (0.793 − 0.793i)5-s + (0.394 + 0.105i)6-s + (0.668 − 0.744i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.396 + 0.687i)10-s + (−0.0721 − 0.269i)11-s − 0.288·12-s + (0.426 − 0.904i)13-s + (−0.320 + 0.630i)14-s + (−0.625 + 0.167i)15-s + (0.125 − 0.216i)16-s + (0.636 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.161 + 0.986i$
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: \((2,\ 546,\ (\ :1/2),\ 0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802110 - 0.681729i\)
\(L(\frac12)\) \(\approx\) \(0.802110 - 0.681729i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 + 1.96i)T \)
13 \( 1 + (-1.53 + 3.26i)T \)
good5 \( 1 + (-1.77 + 1.77i)T - 5iT^{2} \)
11 \( 1 + (0.239 + 0.893i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.62 - 4.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.76 + 0.472i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.27 + 3.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.15 - 5.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.75 + 3.75i)T - 31iT^{2} \)
37 \( 1 + (0.542 + 2.02i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.84 + 6.90i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.61 + 4.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.17 + 2.17i)T + 47iT^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + (-1.46 + 5.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.63 + 2.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.93 + 1.32i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.60 - 13.4i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.41 - 4.41i)T + 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (1.26 - 1.26i)T - 83iT^{2} \)
89 \( 1 + (7.69 - 2.06i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.07 + 0.556i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.50496763205194382040179822897, −9.893458343536745048682202418995, −8.602521123659108910869461141880, −8.124933304775342289663333609420, −7.07039388331347330937616206575, −5.89363393272473794939895904108, −5.37294567166616083719420020910, −3.96874715915082932355390930953, −1.95368117229648327677363690415, −0.858940505412669833513954545434, 1.71312820164401865937179590716, 2.80573639115876480153551353017, 4.40458260329591399444704623206, 5.69174309306835133612961720745, 6.37762939670899639326405656458, 7.43531241681849403528052427638, 8.466499735745204047621547113991, 9.567712355314831839102947734438, 9.941159203894331722771057763020, 11.00442328362528718247806800791

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