# Properties

 Label 2-546-273.32-c1-0-34 Degree $2$ Conductor $546$ Sign $-0.891 - 0.452i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)2-s + (−0.315 − 1.70i)3-s + 1.00i·4-s + (0.258 − 0.965i)5-s + (−0.981 + 1.42i)6-s + (−0.443 − 2.60i)7-s + (0.707 − 0.707i)8-s + (−2.80 + 1.07i)9-s + (−0.865 + 0.499i)10-s + (−2.62 − 0.704i)11-s + (1.70 − 0.315i)12-s + (3.59 − 0.238i)13-s + (−1.53 + 2.15i)14-s + (−1.72 − 0.136i)15-s − 1.00·16-s − 2.39·17-s + ⋯
 L(s)  = 1 + (−0.499 − 0.499i)2-s + (−0.182 − 0.983i)3-s + 0.500i·4-s + (0.115 − 0.431i)5-s + (−0.400 + 0.582i)6-s + (−0.167 − 0.985i)7-s + (0.250 − 0.250i)8-s + (−0.933 + 0.357i)9-s + (−0.273 + 0.158i)10-s + (−0.792 − 0.212i)11-s + (0.491 − 0.0910i)12-s + (0.997 − 0.0661i)13-s + (−0.409 + 0.576i)14-s + (−0.445 − 0.0351i)15-s − 0.250·16-s − 0.580·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.891 - 0.452i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (305, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.891 - 0.452i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.146315 + 0.611537i$$ $$L(\frac12)$$ $$\approx$$ $$0.146315 + 0.611537i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 + 0.707i)T$$
3 $$1 + (0.315 + 1.70i)T$$
7 $$1 + (0.443 + 2.60i)T$$
13 $$1 + (-3.59 + 0.238i)T$$
good5 $$1 + (-0.258 + 0.965i)T + (-4.33 - 2.5i)T^{2}$$
11 $$1 + (2.62 + 0.704i)T + (9.52 + 5.5i)T^{2}$$
17 $$1 + 2.39T + 17T^{2}$$
19 $$1 + (0.807 + 3.01i)T + (-16.4 + 9.5i)T^{2}$$
23 $$1 + 6.02T + 23T^{2}$$
29 $$1 + (2.49 + 1.43i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (0.354 + 1.32i)T + (-26.8 + 15.5i)T^{2}$$
37 $$1 + (-0.770 - 0.770i)T + 37iT^{2}$$
41 $$1 + (-0.406 + 0.108i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (3.05 - 1.76i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (0.717 + 0.192i)T + (40.7 + 23.5i)T^{2}$$
53 $$1 + (-2.12 - 1.22i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (7.69 - 7.69i)T - 59iT^{2}$$
61 $$1 + (0.903 - 1.56i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (0.116 + 0.0311i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 + (-3.68 + 13.7i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (-4.19 + 1.12i)T + (63.2 - 36.5i)T^{2}$$
79 $$1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-7.50 - 7.50i)T + 83iT^{2}$$
89 $$1 + (-7.22 + 7.22i)T - 89iT^{2}$$
97 $$1 + (4.73 + 1.26i)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

−10.58593273192589392273118346881, −9.353578476922765623478424329550, −8.423841339757570582679783765163, −7.74540996037246625930003918103, −6.81609652906684209842035003616, −5.84136600736748458840428955279, −4.47804738720902255582868604014, −3.12402180251519430286391874536, −1.73837403745179431476580199810, −0.42551733841478773088251845829, 2.31288397006085053714716797173, 3.65751505066157277789853752089, 4.98586050842772735836083381680, 5.88149261406093591061210498867, 6.54290969994329234545311564145, 8.038685600386251117761565011479, 8.686169461121446231831675391023, 9.542738084282700923710077593202, 10.35407041690237269126354935612, 10.97495893373480264335346313667