# Properties

 Label 2-273-91.41-c1-0-11 Degree $2$ Conductor $273$ Sign $-0.422 + 0.906i$ Analytic cond. $2.17991$ Root an. cond. $1.47645$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.38 − 0.369i)2-s + (0.866 − 0.5i)3-s + (0.0377 + 0.0217i)4-s + (0.512 + 0.512i)5-s + (−1.38 + 0.369i)6-s + (−2.41 − 1.09i)7-s + (1.97 + 1.97i)8-s + (0.499 − 0.866i)9-s + (−0.517 − 0.896i)10-s + (1.38 − 5.18i)11-s + 0.0435·12-s + (−0.0545 − 3.60i)13-s + (2.92 + 2.39i)14-s + (0.699 + 0.187i)15-s + (−2.04 − 3.53i)16-s + (−1.31 + 2.28i)17-s + ⋯
 L(s)  = 1 + (−0.976 − 0.261i)2-s + (0.499 − 0.288i)3-s + (0.0188 + 0.0108i)4-s + (0.229 + 0.229i)5-s + (−0.563 + 0.151i)6-s + (−0.910 − 0.412i)7-s + (0.699 + 0.699i)8-s + (0.166 − 0.288i)9-s + (−0.163 − 0.283i)10-s + (0.419 − 1.56i)11-s + 0.0125·12-s + (−0.0151 − 0.999i)13-s + (0.781 + 0.641i)14-s + (0.180 + 0.0484i)15-s + (−0.510 − 0.884i)16-s + (−0.319 + 0.553i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $-0.422 + 0.906i$ Analytic conductor: $$2.17991$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{273} (223, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 273,\ (\ :1/2),\ -0.422 + 0.906i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.370450 - 0.581253i$$ $$L(\frac12)$$ $$\approx$$ $$0.370450 - 0.581253i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.866 + 0.5i)T$$
7 $$1 + (2.41 + 1.09i)T$$
13 $$1 + (0.0545 + 3.60i)T$$
good2 $$1 + (1.38 + 0.369i)T + (1.73 + i)T^{2}$$
5 $$1 + (-0.512 - 0.512i)T + 5iT^{2}$$
11 $$1 + (-1.38 + 5.18i)T + (-9.52 - 5.5i)T^{2}$$
17 $$1 + (1.31 - 2.28i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (5.26 - 1.41i)T + (16.4 - 9.5i)T^{2}$$
23 $$1 + (-5.51 + 3.18i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (-0.300 - 0.520i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (6.22 + 6.22i)T + 31iT^{2}$$
37 $$1 + (-0.172 + 0.644i)T + (-32.0 - 18.5i)T^{2}$$
41 $$1 + (2.11 - 7.88i)T + (-35.5 - 20.5i)T^{2}$$
43 $$1 + (-4.10 - 2.36i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (-4.25 + 4.25i)T - 47iT^{2}$$
53 $$1 - 0.282T + 53T^{2}$$
59 $$1 + (1.21 + 4.54i)T + (-51.0 + 29.5i)T^{2}$$
61 $$1 + (-13.0 - 7.55i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-4.48 - 1.20i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 + (-4.11 - 15.3i)T + (-61.4 + 35.5i)T^{2}$$
73 $$1 + (-3.04 + 3.04i)T - 73iT^{2}$$
79 $$1 - 4.77T + 79T^{2}$$
83 $$1 + (-2.42 - 2.42i)T + 83iT^{2}$$
89 $$1 + (5.75 + 1.54i)T + (77.0 + 44.5i)T^{2}$$
97 $$1 + (15.7 - 4.21i)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

−11.12770812108983935951527330562, −10.55204338787701425323701660291, −9.659705154949201108774680191952, −8.675999876484197709813828858365, −8.148783776387827258801987917782, −6.78256745833195086964460140808, −5.79941903465750468501223761369, −3.92964565829740952421758317514, −2.60069351030331694382364629880, −0.69924064443477109624551515402, 1.98966959376635457450061253763, 3.80902898031688993387746657762, 4.93960984078401077555748948761, 6.80111512815170285031012635680, 7.26515494805789726786515843676, 8.890825677778257407968424996205, 9.155671869169116527578807896000, 9.825989030255255062127044628608, 10.92753071423266518824179165956, 12.40624590500927649063623455445