# Properties

 Label 2-546-273.257-c1-0-3 Degree $2$ Conductor $546$ Sign $-0.950 - 0.309i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−1.36 + 1.06i)3-s + 4-s + (2.60 + 1.50i)5-s + (1.36 − 1.06i)6-s + (−2.60 + 0.446i)7-s − 8-s + (0.718 − 2.91i)9-s + (−2.60 − 1.50i)10-s + (−2.01 + 3.49i)11-s + (−1.36 + 1.06i)12-s + (0.138 + 3.60i)13-s + (2.60 − 0.446i)14-s + (−5.15 + 0.730i)15-s + 16-s + 3.34·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.787 + 0.616i)3-s + 0.5·4-s + (1.16 + 0.672i)5-s + (0.556 − 0.436i)6-s + (−0.985 + 0.168i)7-s − 0.353·8-s + (0.239 − 0.970i)9-s + (−0.823 − 0.475i)10-s + (−0.607 + 1.05i)11-s + (−0.393 + 0.308i)12-s + (0.0382 + 0.999i)13-s + (0.696 − 0.119i)14-s + (−1.33 + 0.188i)15-s + 0.250·16-s + 0.810·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.950 - 0.309i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.950 - 0.309i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0836516 + 0.527797i$$ $$L(\frac12)$$ $$\approx$$ $$0.0836516 + 0.527797i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (1.36 - 1.06i)T$$
7 $$1 + (2.60 - 0.446i)T$$
13 $$1 + (-0.138 - 3.60i)T$$
good5 $$1 + (-2.60 - 1.50i)T + (2.5 + 4.33i)T^{2}$$
11 $$1 + (2.01 - 3.49i)T + (-5.5 - 9.52i)T^{2}$$
17 $$1 - 3.34T + 17T^{2}$$
19 $$1 + (2.56 + 4.44i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 - 2.48iT - 23T^{2}$$
29 $$1 + (3.74 - 2.16i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (2.95 + 5.12i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 - 2.57iT - 37T^{2}$$
41 $$1 + (6.29 - 3.63i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (2.23 - 3.87i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (5.77 + 3.33i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (8.50 - 4.90i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 - 3.18iT - 59T^{2}$$
61 $$1 + (5.77 - 3.33i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-7.04 - 4.07i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (0.436 - 0.756i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (6.78 + 11.7i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-8.50 + 14.7i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 12.6iT - 83T^{2}$$
89 $$1 - 3.63iT - 89T^{2}$$
97 $$1 + (-1.10 + 1.91i)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

−10.91570730492295834407324378724, −10.07035396166379175640962221112, −9.687177484382724144102595727159, −9.064561380293154662193551026266, −7.35260360827283126343621118684, −6.57513394290048172090160284434, −5.94521061106871014893203715057, −4.81016800694075466036124514441, −3.25014157664076462233259825967, −1.96369701563219955078555502350, 0.40670245530500768377774995086, 1.77842534595599899403215328522, 3.24819542847674202828093459125, 5.34937329876515984237134013493, 5.81751285065096851596472708569, 6.61599491558190457685435177308, 7.82983748641943281456159888170, 8.572471003884780904361041088709, 9.742981565797192711664955172721, 10.31579911223831364775302579211