# Properties

 Label 1-273-273.107-r1-0-0 Degree $1$ Conductor $273$ Sign $-0.981 + 0.190i$ Analytic cond. $29.3379$ Root an. cond. $29.3379$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 16-s − 17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s − 32-s + ⋯
 L(s)  = 1 − 2-s + 4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 16-s − 17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s − 32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $-0.981 + 0.190i$ Analytic conductor: $$29.3379$$ Root analytic conductor: $$29.3379$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{273} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 273,\ (1:\ ),\ -0.981 + 0.190i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.05596830829 + 0.5811735144i$$ $$L(\frac12)$$ $$\approx$$ $$0.05596830829 + 0.5811735144i$$ $$L(1)$$ $$\approx$$ $$0.6227778316 + 0.2144214012i$$ $$L(1)$$ $$\approx$$ $$0.6227778316 + 0.2144214012i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
13 $$1$$
good2 $$1 - T$$
5 $$1 + (0.5 + 0.866i)T$$
11 $$1 + (0.5 + 0.866i)T$$
17 $$1 - T$$
19 $$1 + (-0.5 + 0.866i)T$$
23 $$1 - T$$
29 $$1 + (0.5 - 0.866i)T$$
31 $$1 + (-0.5 + 0.866i)T$$
37 $$1 + T$$
41 $$1 + (0.5 - 0.866i)T$$
43 $$1 + (-0.5 - 0.866i)T$$
47 $$1 + (0.5 + 0.866i)T$$
53 $$1 + (0.5 - 0.866i)T$$
59 $$1 - T$$
61 $$1 + (-0.5 + 0.866i)T$$
67 $$1 + (-0.5 - 0.866i)T$$
71 $$1 + (0.5 + 0.866i)T$$
73 $$1 + (-0.5 + 0.866i)T$$
79 $$1 + (-0.5 - 0.866i)T$$
83 $$1 - T$$
89 $$1 - T$$
97 $$1 + (-0.5 - 0.866i)T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−25.082975802395893728946303025809, −24.37544866307906183691575350681, −23.70166428234352515109554532730, −21.89091280966704565256563285380, −21.43114326424526832752134561277, −20.02698683031440979449950626695, −19.85700950776515259411936029047, −18.473758670974486290883561995128, −17.68775574912246694166718245986, −16.78062361480597135622600622336, −16.17112915455972105928364797532, −15.08166779295204509690437827036, −13.77658138926850863926348028935, −12.759874838867596075151135457, −11.63088092534190044772193734452, −10.76806211715859308193448597842, −9.53686987859165478609346779462, −8.871855127644025403799387120086, −8.03520615617222455689369618492, −6.61956181816379746364276431945, −5.80189142770056707891918001227, −4.320291957795171835338925108750, −2.66207672262993431778816640029, −1.44766661926720139169172158397, −0.24122640639059654399964346160, 1.67776912565855837974280303802, 2.5361218734772927818316866765, 4.0311766813262257963901828446, 5.89171218002276868105923733817, 6.70006704264767805235981649736, 7.60884957007114834297952552670, 8.83092607476808127635986294516, 9.84724170866970773820516948778, 10.51275600528565966119251985201, 11.5453977036981293487743060311, 12.55836660962568038906668819356, 14.02414092958712568202381988535, 14.9229349130929600486381779089, 15.775065749557453157839597179708, 17.00000914550441480059369901024, 17.74324900548110056425780447305, 18.40649948081991806861162836276, 19.42498826894750866698197990891, 20.21929104932427196391759766951, 21.2683985992281807527473330, 22.179128796667823236596917930823, 23.18216604399375264287293132067, 24.4194992893171688906416574502, 25.33784959892097278091778028329, 25.862929743784230332465419300341