# Properties

 Label 2-405-45.29-c0-0-0 Degree $2$ Conductor $405$ Sign $-0.342 - 0.939i$ Analytic cond. $0.202121$ Root an. cond. $0.449579$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s − 8-s − 0.999·10-s + (0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (0.5 − 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (1 − 1.73i)47-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s − 8-s − 0.999·10-s + (0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (0.5 − 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (1 − 1.73i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $-0.342 - 0.939i$ Analytic conductor: $$0.202121$$ Root analytic conductor: $$0.449579$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{405} (269, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :0),\ -0.342 - 0.939i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7040183938$$ $$L(\frac12)$$ $$\approx$$ $$0.7040183938$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 + (-0.5 - 0.866i)T$$
good2 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
7 $$1 + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.5 + 0.866i)T^{2}$$
13 $$1 + (0.5 - 0.866i)T^{2}$$
17 $$1 - T + T^{2}$$
19 $$1 + T + T^{2}$$
23 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
29 $$1 + (0.5 + 0.866i)T^{2}$$
31 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + (0.5 - 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}$$
53 $$1 - T + T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
83 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
89 $$1 - T^{2}$$
97 $$1 + (0.5 + 0.866i)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

−11.75592060411583780117775129241, −10.58023505175654308698770872629, −9.926452438481325881254636165547, −8.803308771221547068513593430479, −8.019681034380590845444153519589, −6.99376370935908539309254717851, −6.38617212887298080680866494517, −5.40367825166858019471337999689, −3.64897770546299677739154530805, −2.41658163365104649263169340771, 1.30758115796568570267697600701, 2.56436313690679554797468672672, 4.09584441689670257259431011088, 5.50020971905860274164625045316, 6.23157570783589567837452934809, 7.79060459310188126284096061338, 8.761493075740315147376689294646, 9.573162261905505329327860118547, 10.17139498500539334332012494454, 11.12919523441986864015023418405