Properties

Label 2-546-273.185-c1-0-6
Degree $2$
Conductor $546$
Sign $-0.628 - 0.777i$
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 + (0.323 + 1.70i)3-s + (0.499 + 0.866i)4-s + (1.99 + 3.45i)5-s + (0.570 − 1.63i)6-s + (−2.44 − 1.01i)7-s − 0.999i·8-s + (−2.79 + 1.10i)9-s − 3.98i·10-s + 3.67i·11-s + (−1.31 + 1.13i)12-s + (3.60 + 0.127i)13-s + (1.61 + 2.09i)14-s + (−5.22 + 4.50i)15-s + (−0.5 + 0.866i)16-s + (1.17 + 2.03i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.186 + 0.982i)3-s + (0.249 + 0.433i)4-s + (0.891 + 1.54i)5-s + (0.232 − 0.667i)6-s + (−0.924 − 0.381i)7-s − 0.353i·8-s + (−0.930 + 0.366i)9-s − 1.26i·10-s + 1.10i·11-s + (−0.378 + 0.326i)12-s + (0.999 + 0.0353i)13-s + (0.430 + 0.560i)14-s + (−1.34 + 1.16i)15-s + (−0.125 + 0.216i)16-s + (0.284 + 0.492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.628 - 0.777i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461466 + 0.965797i\)
\(L(\frac12)\) \(\approx\) \(0.461466 + 0.965797i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.323 - 1.70i)T \)
7 \( 1 + (2.44 + 1.01i)T \)
13 \( 1 + (-3.60 - 0.127i)T \)
good5 \( 1 + (-1.99 - 3.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.67iT - 11T^{2} \)
17 \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 4.20iT - 19T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.39 + 1.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.42 + 4.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.44 - 5.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.55 + 2.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.70 + 4.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.81 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.81 + 1.04i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.85 - 8.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.276iT - 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + (3.24 + 1.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.474 + 0.273i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 - 2.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-0.959 + 1.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.07 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.68836082086669970647207212008, −10.23696572633554570084595801547, −9.648078486560011060890776698494, −8.872680342054506472226315554012, −7.47989990594184557947325391267, −6.63247452320439588383180747918, −5.80124052426843946653907134361, −4.09596402425079031794807689000, −3.17655181177889968426289369435, −2.23716564906679455343202613405, 0.73601320576156413579975793892, 1.85633592005399976266331132233, 3.43828149997298651206234775659, 5.54608259105562893664854483683, 5.79485673738015643525186062812, 6.78780370899795979148138040499, 8.114171577254717626134306186377, 8.737561492557301934851304406410, 9.201818634010600953235216599085, 10.20691440338306095593680108229

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