# Properties

 Label 2-3645-135.29-c0-0-5 Degree $2$ Conductor $3645$ Sign $-0.230 + 0.973i$ Analytic cond. $1.81909$ Root an. cond. $1.34873$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.173 − 0.984i)2-s + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (−0.939 + 0.342i)34-s + (−0.766 − 0.642i)38-s + (0.173 + 0.984i)40-s + (−0.5 + 0.866i)46-s + (1.87 − 0.684i)47-s + ⋯
 L(s)  = 1 + (0.173 − 0.984i)2-s + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (0.939 + 0.342i)31-s + (−0.939 + 0.342i)34-s + (−0.766 − 0.642i)38-s + (0.173 + 0.984i)40-s + (−0.5 + 0.866i)46-s + (1.87 − 0.684i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3645$$    =    $$3^{6} \cdot 5$$ Sign: $-0.230 + 0.973i$ Analytic conductor: $$1.81909$$ Root analytic conductor: $$1.34873$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3645} (809, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3645,\ (\ :0),\ -0.230 + 0.973i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.319605918$$ $$L(\frac12)$$ $$\approx$$ $$1.319605918$$ $$L(1)$$ 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 + (0.766 - 0.642i)T$$
good2 $$1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}$$
7 $$1 + (-0.766 + 0.642i)T^{2}$$
11 $$1 + (-0.173 - 0.984i)T^{2}$$
13 $$1 + (0.939 - 0.342i)T^{2}$$
17 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}$$
29 $$1 + (0.939 + 0.342i)T^{2}$$
31 $$1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}$$
37 $$1 + (0.5 - 0.866i)T^{2}$$
41 $$1 + (0.939 - 0.342i)T^{2}$$
43 $$1 + (-0.173 - 0.984i)T^{2}$$
47 $$1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2}$$
53 $$1 - T + T^{2}$$
59 $$1 + (-0.173 + 0.984i)T^{2}$$
61 $$1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}$$
67 $$1 + (0.939 - 0.342i)T^{2}$$
71 $$1 + (0.5 - 0.866i)T^{2}$$
73 $$1 + (0.5 + 0.866i)T^{2}$$
79 $$1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}$$
83 $$1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + (-0.173 - 0.984i)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

−8.553444606826684461377066928423, −7.66307921960416278433821479135, −7.02075148659398175393574135228, −6.54108918756878526537999998708, −5.30485592085633938626254478723, −4.31750946943726764677477736109, −3.77420088841146591466932320896, −2.77139293116896321575422762839, −2.33101783473370229030080001275, −0.77014015216994414075940004411, 1.34595772554037149266735576167, 2.50627194864857714700282749014, 3.90032189161582702662827503091, 4.34898727848242622195522475595, 5.43055839240802972604119706458, 5.87542447470779119470381053979, 6.74452797543492343788559316282, 7.53523803557178324032535801589, 8.038933313102685840631632451928, 8.579496642142696082179946638142