# Properties

 Label 2-546-273.32-c1-0-4 Degree $2$ Conductor $546$ Sign $-0.843 - 0.537i$ 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 + (−1.73 − 0.00254i)3-s + 1.00i·4-s + (−0.852 + 3.18i)5-s + (−1.22 − 1.22i)6-s + (2.55 − 0.680i)7-s + (−0.707 + 0.707i)8-s + (2.99 + 0.00880i)9-s + (−2.85 + 1.64i)10-s + (4.19 + 1.12i)11-s + (0.00254 − 1.73i)12-s + (−3.59 + 0.221i)13-s + (2.28 + 1.32i)14-s + (1.48 − 5.50i)15-s − 1.00·16-s − 5.29·17-s + ⋯
 L(s)  = 1 + (0.499 + 0.499i)2-s + (−0.999 − 0.00146i)3-s + 0.500i·4-s + (−0.381 + 1.42i)5-s + (−0.499 − 0.500i)6-s + (0.966 − 0.257i)7-s + (−0.250 + 0.250i)8-s + (0.999 + 0.00293i)9-s + (−0.902 + 0.520i)10-s + (1.26 + 0.338i)11-s + (0.000734 − 0.499i)12-s + (−0.998 + 0.0615i)13-s + (0.611 + 0.354i)14-s + (0.383 − 1.42i)15-s − 0.250·16-s − 1.28·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.843 - 0.537i$ 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: $$(2,\ 546,\ (\ :1/2),\ -0.843 - 0.537i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.335709 + 1.15175i$$ $$L(\frac12)$$ $$\approx$$ $$0.335709 + 1.15175i$$ $$L(\frac{3}{2})$$ 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 + (1.73 + 0.00254i)T$$
7 $$1 + (-2.55 + 0.680i)T$$
13 $$1 + (3.59 - 0.221i)T$$
good5 $$1 + (0.852 - 3.18i)T + (-4.33 - 2.5i)T^{2}$$
11 $$1 + (-4.19 - 1.12i)T + (9.52 + 5.5i)T^{2}$$
17 $$1 + 5.29T + 17T^{2}$$
19 $$1 + (-1.57 - 5.88i)T + (-16.4 + 9.5i)T^{2}$$
23 $$1 + 8.97T + 23T^{2}$$
29 $$1 + (-2.35 - 1.36i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (0.298 + 1.11i)T + (-26.8 + 15.5i)T^{2}$$
37 $$1 + (-1.48 - 1.48i)T + 37iT^{2}$$
41 $$1 + (-8.93 + 2.39i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (2.42 - 1.40i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (-4.19 - 1.12i)T + (40.7 + 23.5i)T^{2}$$
53 $$1 + (4.94 + 2.85i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (0.571 - 0.571i)T - 59iT^{2}$$
61 $$1 + (-1.18 + 2.05i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-9.45 - 2.53i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 + (2.23 - 8.35i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (1.45 - 0.390i)T + (63.2 - 36.5i)T^{2}$$
79 $$1 + (0.257 + 0.446i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-4.56 - 4.56i)T + 83iT^{2}$$
89 $$1 + (1.56 - 1.56i)T - 89iT^{2}$$
97 $$1 + (-2.82 - 0.756i)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

−11.38337455206141779255763846096, −10.49261665444569150211694913529, −9.654123853554287559175636875669, −8.054601167276961843833176896558, −7.27202684775332948985168869859, −6.62770566980173129318339266693, −5.81265656593731645566989128924, −4.47484490829903129483394239333, −3.89547029069581584466471324133, −2.05749625646383380939001377426, 0.68222530391863990908291857860, 1.99146599022320517518822743639, 4.27961450674273842188810319675, 4.55960545724293033473377012042, 5.49109054736824135047489074304, 6.52575649804428896596565231340, 7.75518991003362629091540760042, 8.899579496146609486168905797601, 9.532527330357955265007793220235, 10.81236835277685936153747830197