# Properties

 Label 2-546-91.34-c1-0-2 Degree $2$ Conductor $546$ Sign $-0.856 + 0.515i$ 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 + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−2.27 + 2.27i)5-s + (−0.707 + 0.707i)6-s + (−2.27 − 1.35i)7-s + (−0.707 + 0.707i)8-s − 9-s − 3.21·10-s + (0.355 − 0.355i)11-s − 1.00·12-s + (−2 − 3i)13-s + (−0.648 − 2.56i)14-s + (−2.27 − 2.27i)15-s − 1.00·16-s + 4.32·17-s + ⋯
 L(s)  = 1 + (0.499 + 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (−1.01 + 1.01i)5-s + (−0.288 + 0.288i)6-s + (−0.858 − 0.512i)7-s + (−0.250 + 0.250i)8-s − 0.333·9-s − 1.01·10-s + (0.107 − 0.107i)11-s − 0.288·12-s + (−0.554 − 0.832i)13-s + (−0.173 − 0.685i)14-s + (−0.586 − 0.586i)15-s − 0.250·16-s + 1.04·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.856 + 0.515i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (307, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.856 + 0.515i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.178308 - 0.642338i$$ $$L(\frac12)$$ $$\approx$$ $$0.178308 - 0.642338i$$ $$L(\frac{3}{2})$$ 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 - iT$$
7 $$1 + (2.27 + 1.35i)T$$
13 $$1 + (2 + 3i)T$$
good5 $$1 + (2.27 - 2.27i)T - 5iT^{2}$$
11 $$1 + (-0.355 + 0.355i)T - 11iT^{2}$$
17 $$1 - 4.32T + 17T^{2}$$
19 $$1 + (5.98 - 5.98i)T - 19iT^{2}$$
23 $$1 + 2.38iT - 23T^{2}$$
29 $$1 - 1.09T + 29T^{2}$$
31 $$1 + (1.08 - 1.08i)T - 31iT^{2}$$
37 $$1 + (5.18 - 5.18i)T - 37iT^{2}$$
41 $$1 + (3.53 - 3.53i)T - 41iT^{2}$$
43 $$1 - 7.44iT - 43T^{2}$$
47 $$1 + (-4.71 - 4.71i)T + 47iT^{2}$$
53 $$1 + 11.2T + 53T^{2}$$
59 $$1 + (3.61 + 3.61i)T + 59iT^{2}$$
61 $$1 + 4.32iT - 61T^{2}$$
67 $$1 + (0.531 + 0.531i)T + 67iT^{2}$$
71 $$1 + (-6.38 - 6.38i)T + 71iT^{2}$$
73 $$1 + (-5.18 - 5.18i)T + 73iT^{2}$$
79 $$1 + 11.3T + 79T^{2}$$
83 $$1 + (6.71 - 6.71i)T - 83iT^{2}$$
89 $$1 + (-3.75 - 3.75i)T + 89iT^{2}$$
97 $$1 + (-12.9 + 12.9i)T - 97iT^{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

−11.22119635801516766509966082828, −10.40592162694200815416269773188, −9.839611261464844290212046206364, −8.317068813277294864814961562888, −7.70514705300580946801869500574, −6.72048394284943791511316786852, −5.94871704967425651551681092484, −4.58071381670826791247292107013, −3.57863053519314764481235666036, −3.04060188597766700051696392052, 0.31261432854847018296460341329, 2.07731796117217031843678152301, 3.44832421261165002470313527916, 4.48424210418940467974105041290, 5.45436000630638271966725043103, 6.62964488132516350036814839764, 7.49513533358234226230251911659, 8.753732384268770195545175088946, 9.188309656977777487441998931651, 10.42511969371927479012082006140