# Properties

 Label 2-1805-5.4-c1-0-124 Degree $2$ Conductor $1805$ Sign $0.223 + 0.974i$ Analytic cond. $14.4129$ Root an. cond. $3.79644$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·4-s + (−0.5 − 2.17i)5-s − 4.35i·7-s + 3·9-s + 5·11-s + 4·16-s − 4.35i·17-s + (−1 − 4.35i)20-s + 8.71i·23-s + (−4.50 + 2.17i)25-s − 8.71i·28-s + (−9.50 + 2.17i)35-s + 6·36-s + 13.0i·43-s + 10·44-s + (−1.5 − 6.53i)45-s + ⋯
 L(s)  = 1 + 4-s + (−0.223 − 0.974i)5-s − 1.64i·7-s + 9-s + 1.50·11-s + 16-s − 1.05i·17-s + (−0.223 − 0.974i)20-s + 1.81i·23-s + (−0.900 + 0.435i)25-s − 1.64i·28-s + (−1.60 + 0.368i)35-s + 36-s + 1.99i·43-s + 1.50·44-s + (−0.223 − 0.974i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1805$$    =    $$5 \cdot 19^{2}$$ Sign: $0.223 + 0.974i$ Analytic conductor: $$14.4129$$ Root analytic conductor: $$3.79644$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1805} (1084, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1805,\ (\ :1/2),\ 0.223 + 0.974i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.571649156$$ $$L(\frac12)$$ $$\approx$$ $$2.571649156$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (0.5 + 2.17i)T$$
19 $$1$$
good2 $$1 - 2T^{2}$$
3 $$1 - 3T^{2}$$
7 $$1 + 4.35iT - 7T^{2}$$
11 $$1 - 5T + 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 4.35iT - 17T^{2}$$
23 $$1 - 8.71iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 - 13.0iT - 43T^{2}$$
47 $$1 + 4.35iT - 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + 15T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 13.0iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 8.71iT - 83T^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 - 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

−9.436385365048427922057813788004, −8.066024701715080433411156361123, −7.32784909456880983184783824028, −7.01185268922805688902751875529, −6.04557888591264506973551174429, −4.81898104525831975047546655384, −4.07479631525240120646021361236, −3.37818609089530100132707616139, −1.52684418489301665659941999425, −1.09138253188100489468037931358, 1.67373409149660737061842670940, 2.40443777649611652640449082062, 3.41206874872959085804217333513, 4.33917776647577494876924816201, 5.75431925473642207948907103346, 6.44590779767483461174758937445, 6.77199392012558742286058165180, 7.79673289714273862146197698551, 8.654291319180622309730142326698, 9.420264481221764838487302267372