# Properties

 Label 2-273-13.12-c1-0-3 Degree $2$ Conductor $273$ Sign $0.342 - 0.939i$ 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 + 0.233i·2-s − 3-s + 1.94·4-s + 2.94i·5-s − 0.233i·6-s − i·7-s + 0.921i·8-s + 9-s − 0.687·10-s + 3.62i·11-s − 1.94·12-s + (−3.38 − 1.23i)13-s + 0.233·14-s − 2.94i·15-s + 3.67·16-s + 1.53·17-s + ⋯
 L(s)  = 1 + 0.165i·2-s − 0.577·3-s + 0.972·4-s + 1.31i·5-s − 0.0953i·6-s − 0.377i·7-s + 0.325i·8-s + 0.333·9-s − 0.217·10-s + 1.09i·11-s − 0.561·12-s + (−0.939 − 0.342i)13-s + 0.0623·14-s − 0.760i·15-s + 0.918·16-s + 0.371·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $0.342 - 0.939i$ Analytic conductor: $$2.17991$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{273} (64, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 273,\ (\ :1/2),\ 0.342 - 0.939i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.04606 + 0.732398i$$ $$L(\frac12)$$ $$\approx$$ $$1.04606 + 0.732398i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
7 $$1 + iT$$
13 $$1 + (3.38 + 1.23i)T$$
good2 $$1 - 0.233iT - 2T^{2}$$
5 $$1 - 2.94iT - 5T^{2}$$
11 $$1 - 3.62iT - 11T^{2}$$
17 $$1 - 1.53T + 17T^{2}$$
19 $$1 - 4.10iT - 19T^{2}$$
23 $$1 - 7.03T + 23T^{2}$$
29 $$1 + 3.79T + 29T^{2}$$
31 $$1 - 1.79iT - 31T^{2}$$
37 $$1 + 7.24iT - 37T^{2}$$
41 $$1 + 9.15iT - 41T^{2}$$
43 $$1 - 5.79T + 43T^{2}$$
47 $$1 + 7.25iT - 47T^{2}$$
53 $$1 + 13.3T + 53T^{2}$$
59 $$1 - 4.84iT - 59T^{2}$$
61 $$1 - 2.77T + 61T^{2}$$
67 $$1 + 14.8iT - 67T^{2}$$
71 $$1 + 3.62iT - 71T^{2}$$
73 $$1 + 14.5iT - 73T^{2}$$
79 $$1 - 6.56T + 79T^{2}$$
83 $$1 - 7.25iT - 83T^{2}$$
89 $$1 - 0.636iT - 89T^{2}$$
97 $$1 - 11.3iT - 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

−12.06606073871970103863441831488, −10.85622517881924753551020875685, −10.58984709492269764374253739628, −9.584338727379922091510486399410, −7.51893974480926736727654416702, −7.27803106750085300850772100744, −6.32028790172054820452278063657, −5.15731459349275679588299606724, −3.44824586229474762633637821250, −2.12247920713668402735662001322, 1.13444297621586274594223265677, 2.89364023055856544602029483310, 4.69741478460604261659142356811, 5.58414330373274965584556389889, 6.65192502676089845684662448688, 7.80809845386472579215069953803, 8.933432722629424654857146771945, 9.816896536178062028906560560627, 11.21323758108477599505358190533, 11.52202572924416195198343244874