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Properties

Label 2-546-91.81-c1-0-12
Degree $2$
Conductor $546$
Sign $0.0710 + 0.997i$
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  + (−0.5 + 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−0.228 − 0.395i)5-s + (0.5 − 0.866i)6-s + (2.45 − 0.989i)7-s + 0.999·8-s + 9-s + 0.456·10-s − 3.83·11-s + (0.499 + 0.866i)12-s + (−3.13 − 1.78i)13-s + (−0.369 + 2.61i)14-s + (0.228 + 0.395i)15-s + (−0.5 + 0.866i)16-s + (−0.775 − 1.34i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.102 − 0.176i)5-s + (0.204 − 0.353i)6-s + (0.927 − 0.374i)7-s + 0.353·8-s + 0.333·9-s + 0.144·10-s − 1.15·11-s + (0.144 + 0.249i)12-s + (−0.869 − 0.494i)13-s + (−0.0988 + 0.700i)14-s + (0.0589 + 0.102i)15-s + (−0.125 + 0.216i)16-s + (−0.188 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0710 + 0.997i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0710 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409145 - 0.381043i\)
\(L(\frac12)\) \(\approx\) \(0.409145 - 0.381043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + T \)
7 \( 1 + (-2.45 + 0.989i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (0.228 + 0.395i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.83T + 11T^{2} \)
17 \( 1 + (0.775 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.88T + 19T^{2} \)
23 \( 1 + (1.62 - 2.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.20 + 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.80 + 8.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.140 - 0.242i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.57 + 6.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.93 + 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.550 + 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.68 - 8.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 1.78T + 67T^{2} \)
71 \( 1 + (5.06 - 8.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.70 - 4.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.35T + 83T^{2} \)
89 \( 1 + (-0.0898 + 0.155i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.73 + 8.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.42667981241988095026546362443, −9.911733082912663556462914442300, −8.564117348256912485306660227558, −7.81882118618381212600430336567, −7.17781508424897512186537002170, −5.90391061718832897152251441572, −5.08309139589809455943514397094, −4.29881591152089533692565540595, −2.26751345546645888168391719033, −0.38157355843925044350891784353, 1.71097432284838726656442326217, 2.91632931095892330012021134408, 4.56928656113881276190019579225, 5.14317915949037776302361700908, 6.52542199693466425334711568823, 7.60848592195704706682714789148, 8.371375774988981117936721177543, 9.349852227338352850655500419641, 10.49814850039436338485848969632, 10.83563268995662351562897258286

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