# Properties

 Label 2-495-55.54-c0-0-1 Degree $2$ Conductor $495$ Sign $1$ Analytic cond. $0.247037$ Root an. cond. $0.497028$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 1.41·2-s + 1.00·4-s + 5-s + 1.41·7-s − 1.41·10-s − 11-s − 1.41·13-s − 2.00·14-s − 0.999·16-s + 1.41·17-s + 1.00·20-s + 1.41·22-s + 25-s + 2.00·26-s + 1.41·28-s + 1.41·32-s − 2.00·34-s + 1.41·35-s − 1.41·43-s − 1.00·44-s + 1.00·49-s − 1.41·50-s − 1.41·52-s − 55-s − 1.00·64-s − 1.41·65-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.00·4-s + 5-s + 1.41·7-s − 1.41·10-s − 11-s − 1.41·13-s − 2.00·14-s − 0.999·16-s + 1.41·17-s + 1.00·20-s + 1.41·22-s + 25-s + 2.00·26-s + 1.41·28-s + 1.41·32-s − 2.00·34-s + 1.41·35-s − 1.41·43-s − 1.00·44-s + 1.00·49-s − 1.41·50-s − 1.41·52-s − 55-s − 1.00·64-s − 1.41·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$495$$    =    $$3^{2} \cdot 5 \cdot 11$$ Sign: $1$ Analytic conductor: $$0.247037$$ Root analytic conductor: $$0.497028$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{495} (109, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 495,\ (\ :0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5618186560$$ $$L(\frac12)$$ $$\approx$$ $$0.5618186560$$ $$L(1)$$ 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 - T$$
11 $$1 + T$$
good2 $$1 + 1.41T + T^{2}$$
7 $$1 - 1.41T + T^{2}$$
13 $$1 + 1.41T + T^{2}$$
17 $$1 - 1.41T + T^{2}$$
19 $$1 - T^{2}$$
23 $$1 - T^{2}$$
29 $$1 - T^{2}$$
31 $$1 + T^{2}$$
37 $$1 - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 + 1.41T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + T^{2}$$
73 $$1 - 1.41T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + 1.41T + T^{2}$$
89 $$1 + 2T + T^{2}$$
97 $$1 - 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

−10.76660777022994655235228169128, −10.08336258337823404344411885192, −9.563289442620246560947755463579, −8.401864268159942187199920064110, −7.83609449669001126086671876843, −7.02468406012674087089570360911, −5.45654632625930193207409868945, −4.82582518091095724664559359195, −2.56324593771758606772819653651, −1.51719217207667209008065103625, 1.51719217207667209008065103625, 2.56324593771758606772819653651, 4.82582518091095724664559359195, 5.45654632625930193207409868945, 7.02468406012674087089570360911, 7.83609449669001126086671876843, 8.401864268159942187199920064110, 9.563289442620246560947755463579, 10.08336258337823404344411885192, 10.76660777022994655235228169128