# Properties

 Label 2-495-1.1-c3-0-33 Degree $2$ Conductor $495$ Sign $-1$ Analytic cond. $29.2059$ Root an. cond. $5.40425$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.56·2-s − 5.56·4-s + 5·5-s − 10.2·7-s + 21.1·8-s − 7.80·10-s + 11·11-s − 40.8·13-s + 16·14-s + 11.4·16-s + 98.7·17-s − 39.6·19-s − 27.8·20-s − 17.1·22-s − 61.6·23-s + 25·25-s + 63.8·26-s + 56.9·28-s + 149.·29-s + 54.7·31-s − 187.·32-s − 154.·34-s − 51.2·35-s + 44.8·37-s + 61.9·38-s + 105.·40-s − 336.·41-s + ⋯
 L(s)  = 1 − 0.552·2-s − 0.695·4-s + 0.447·5-s − 0.553·7-s + 0.935·8-s − 0.246·10-s + 0.301·11-s − 0.872·13-s + 0.305·14-s + 0.178·16-s + 1.40·17-s − 0.478·19-s − 0.310·20-s − 0.166·22-s − 0.559·23-s + 0.200·25-s + 0.481·26-s + 0.384·28-s + 0.954·29-s + 0.317·31-s − 1.03·32-s − 0.777·34-s − 0.247·35-s + 0.199·37-s + 0.264·38-s + 0.418·40-s − 1.28·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$495$$    =    $$3^{2} \cdot 5 \cdot 11$$ Sign: $-1$ Analytic conductor: $$29.2059$$ Root analytic conductor: $$5.40425$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 495,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 5T$$
11 $$1 - 11T$$
good2 $$1 + 1.56T + 8T^{2}$$
7 $$1 + 10.2T + 343T^{2}$$
13 $$1 + 40.8T + 2.19e3T^{2}$$
17 $$1 - 98.7T + 4.91e3T^{2}$$
19 $$1 + 39.6T + 6.85e3T^{2}$$
23 $$1 + 61.6T + 1.21e4T^{2}$$
29 $$1 - 149.T + 2.43e4T^{2}$$
31 $$1 - 54.7T + 2.97e4T^{2}$$
37 $$1 - 44.8T + 5.06e4T^{2}$$
41 $$1 + 336.T + 6.89e4T^{2}$$
43 $$1 + 2.36T + 7.95e4T^{2}$$
47 $$1 - 333.T + 1.03e5T^{2}$$
53 $$1 + 640.T + 1.48e5T^{2}$$
59 $$1 - 370.T + 2.05e5T^{2}$$
61 $$1 + 714.T + 2.26e5T^{2}$$
67 $$1 + 404.T + 3.00e5T^{2}$$
71 $$1 + 939.T + 3.57e5T^{2}$$
73 $$1 + 362.T + 3.89e5T^{2}$$
79 $$1 - 951.T + 4.93e5T^{2}$$
83 $$1 + 735.T + 5.71e5T^{2}$$
89 $$1 + 385.T + 7.04e5T^{2}$$
97 $$1 + 966.T + 9.12e5T^{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

−9.948163573622847215252595845269, −9.371755726495176599387817081107, −8.383063868392147302400508649267, −7.54440657040370455372181697887, −6.41797549725975424547959293615, −5.33945680527724516925475748585, −4.31975899509896448404227576913, −3.01925994101984347803007631576, −1.41939565351603278287198016777, 0, 1.41939565351603278287198016777, 3.01925994101984347803007631576, 4.31975899509896448404227576913, 5.33945680527724516925475748585, 6.41797549725975424547959293615, 7.54440657040370455372181697887, 8.383063868392147302400508649267, 9.371755726495176599387817081107, 9.948163573622847215252595845269