# Properties

 Label 2-546-13.12-c1-0-6 Degree $2$ Conductor $546$ Sign $0.155 + 0.987i$ 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 − i·2-s − 3-s − 4-s + 0.561i·5-s + i·6-s + i·7-s + i·8-s + 9-s + 0.561·10-s − 1.43i·11-s + 12-s + (−0.561 − 3.56i)13-s + 14-s − 0.561i·15-s + 16-s + 5.68·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.251i·5-s + 0.408i·6-s + 0.377i·7-s + 0.353i·8-s + 0.333·9-s + 0.177·10-s − 0.433i·11-s + 0.288·12-s + (−0.155 − 0.987i)13-s + 0.267·14-s − 0.144i·15-s + 0.250·16-s + 1.37·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.155 + 0.987i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.155 + 0.987i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.846329 - 0.723343i$$ $$L(\frac12)$$ $$\approx$$ $$0.846329 - 0.723343i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + T$$
7 $$1 - iT$$
13 $$1 + (0.561 + 3.56i)T$$
good5 $$1 - 0.561iT - 5T^{2}$$
11 $$1 + 1.43iT - 11T^{2}$$
17 $$1 - 5.68T + 17T^{2}$$
19 $$1 + 2.56iT - 19T^{2}$$
23 $$1 - 5.68T + 23T^{2}$$
29 $$1 + 2.56T + 29T^{2}$$
31 $$1 + 10.2iT - 31T^{2}$$
37 $$1 - 1.68iT - 37T^{2}$$
41 $$1 + 4iT - 41T^{2}$$
43 $$1 + 10.5T + 43T^{2}$$
47 $$1 + 6.24iT - 47T^{2}$$
53 $$1 - 13.1T + 53T^{2}$$
59 $$1 - 12.2iT - 59T^{2}$$
61 $$1 - 2.56T + 61T^{2}$$
67 $$1 + 7.12iT - 67T^{2}$$
71 $$1 - 15.3iT - 71T^{2}$$
73 $$1 + 7.43iT - 73T^{2}$$
79 $$1 - 16T + 79T^{2}$$
83 $$1 - 2iT - 83T^{2}$$
89 $$1 - 8iT - 89T^{2}$$
97 $$1 + 10iT - 97T^{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

−10.65293478224083194080861743413, −9.989144793989666392904901866589, −9.010592278323758335246929041063, −8.010271066408051942881125907744, −6.97364234987324441591874111064, −5.69016212473853088460661458379, −5.10285295172301647332546109227, −3.63539967962175318920647713713, −2.62615273599389016373788112315, −0.834707819000932310465833839667, 1.31355843075987360880619276384, 3.46453253255970694580335035908, 4.71893763562569546200998891954, 5.36164933279997614873877654856, 6.61875524705064131753866320282, 7.17424229232181877757388618748, 8.223207276587349189906413582017, 9.233520644683733315640114958847, 10.04087424549474826959752052262, 10.91252293023656831853274305266