# Properties

 Label 2-546-13.10-c1-0-3 Degree $2$ Conductor $546$ Sign $0.667 - 0.744i$ 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 + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − 0.332i·5-s + (−0.866 − 0.499i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.166 + 0.288i)10-s + (2.26 − 1.30i)11-s + 0.999·12-s + (3.41 − 1.16i)13-s + 0.999·14-s + (0.288 − 0.166i)15-s + (−0.5 − 0.866i)16-s + (−1.94 + 3.36i)17-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.148i·5-s + (−0.353 − 0.204i)6-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0526 + 0.0911i)10-s + (0.681 − 0.393i)11-s + 0.288·12-s + (0.946 − 0.323i)13-s + 0.267·14-s + (0.0744 − 0.0429i)15-s + (−0.125 − 0.216i)16-s + (−0.471 + 0.816i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.667 - 0.744i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.667 - 0.744i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.13165 + 0.505660i$$ $$L(\frac12)$$ $$\approx$$ $$1.13165 + 0.505660i$$ $$L(\frac{3}{2})$$ 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 + (-0.5 - 0.866i)T$$
7 $$1 + (0.866 + 0.5i)T$$
13 $$1 + (-3.41 + 1.16i)T$$
good5 $$1 + 0.332iT - 5T^{2}$$
11 $$1 + (-2.26 + 1.30i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-4.85 - 2.80i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-2.10 - 3.64i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-0.593 - 1.02i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 7.07iT - 31T^{2}$$
37 $$1 + (-0.499 + 0.288i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 + (-0.451 + 0.260i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-1.53 + 2.66i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 - 12.0iT - 47T^{2}$$
53 $$1 + 9.71T + 53T^{2}$$
59 $$1 + (-9.07 - 5.23i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (3.71 - 6.44i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-10.3 + 5.96i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (0.818 + 0.472i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 - 4.66iT - 73T^{2}$$
79 $$1 + 0.943T + 79T^{2}$$
83 $$1 + 13.7iT - 83T^{2}$$
89 $$1 + (-2.92 + 1.69i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (5.03 + 2.90i)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

−10.83355609530452157101228701212, −9.839699141875592161504024291142, −9.144509710331623613908121675418, −8.400953944253608679913008887350, −7.50268761316838694964059406810, −6.34658706230233626192044419964, −5.56977303727984322204221486427, −4.15570252696066319005374244641, −3.14060414376862508795376117042, −1.27720965018775831276361634425, 1.09057123630961569093770172167, 2.54695127516940982732816701043, 3.57077956221773455200562312936, 5.02033403673555551678467121960, 6.67223679115452552462770118025, 6.88884640513958376694074075719, 8.207444072067814404740625425372, 9.015570408264289140388369077208, 9.567062965184724702666482549688, 10.75164477818556914080396249984