# Properties

 Label 2-45e2-1.1-c1-0-53 Degree $2$ Conductor $2025$ Sign $-1$ Analytic cond. $16.1697$ Root an. cond. $4.02115$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.473·2-s − 1.77·4-s − 2.56·7-s − 1.78·8-s + 6.16·11-s − 2.13·13-s − 1.21·14-s + 2.70·16-s + 3.16·17-s + 0.356·19-s + 2.91·22-s − 4.21·23-s − 1.00·26-s + 4.55·28-s − 1.68·29-s − 8.25·31-s + 4.85·32-s + 1.49·34-s − 3.63·37-s + 0.168·38-s + 2.73·41-s + 7.67·43-s − 10.9·44-s − 1.99·46-s − 11.4·47-s − 0.430·49-s + 3.78·52-s + ⋯
 L(s)  = 1 + 0.334·2-s − 0.888·4-s − 0.968·7-s − 0.631·8-s + 1.85·11-s − 0.591·13-s − 0.324·14-s + 0.676·16-s + 0.768·17-s + 0.0817·19-s + 0.622·22-s − 0.878·23-s − 0.197·26-s + 0.860·28-s − 0.313·29-s − 1.48·31-s + 0.858·32-s + 0.257·34-s − 0.597·37-s + 0.0273·38-s + 0.426·41-s + 1.17·43-s − 1.65·44-s − 0.293·46-s − 1.66·47-s − 0.0615·49-s + 0.525·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2025$$    =    $$3^{4} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$16.1697$$ Root analytic conductor: $$4.02115$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2025,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 - 0.473T + 2T^{2}$$
7 $$1 + 2.56T + 7T^{2}$$
11 $$1 - 6.16T + 11T^{2}$$
13 $$1 + 2.13T + 13T^{2}$$
17 $$1 - 3.16T + 17T^{2}$$
19 $$1 - 0.356T + 19T^{2}$$
23 $$1 + 4.21T + 23T^{2}$$
29 $$1 + 1.68T + 29T^{2}$$
31 $$1 + 8.25T + 31T^{2}$$
37 $$1 + 3.63T + 37T^{2}$$
41 $$1 - 2.73T + 41T^{2}$$
43 $$1 - 7.67T + 43T^{2}$$
47 $$1 + 11.4T + 47T^{2}$$
53 $$1 + 9.43T + 53T^{2}$$
59 $$1 + 10.2T + 59T^{2}$$
61 $$1 + 0.0109T + 61T^{2}$$
67 $$1 + 0.982T + 67T^{2}$$
71 $$1 - 6.43T + 71T^{2}$$
73 $$1 + 6.61T + 73T^{2}$$
79 $$1 + 9.47T + 79T^{2}$$
83 $$1 + 10.4T + 83T^{2}$$
89 $$1 - 6.26T + 89T^{2}$$
97 $$1 + 7.20T + 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

−9.023081920991468464721698896793, −8.025022495450521621025491735309, −7.08683558838730343538509206153, −6.24082805245264914836434530362, −5.62797907280854838367637595226, −4.53301660238679966520656431098, −3.77054734760084612839955585112, −3.16037619216315454312562349622, −1.51586454388152440442471438756, 0, 1.51586454388152440442471438756, 3.16037619216315454312562349622, 3.77054734760084612839955585112, 4.53301660238679966520656431098, 5.62797907280854838367637595226, 6.24082805245264914836434530362, 7.08683558838730343538509206153, 8.025022495450521621025491735309, 9.023081920991468464721698896793