# Properties

 Degree $2$ Conductor $441$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2·4-s − 7·13-s + 4·16-s − 7·19-s − 5·25-s − 7·31-s − 37-s + 5·43-s + 14·52-s + 14·61-s − 8·64-s + 11·67-s − 7·73-s + 14·76-s − 13·79-s + 14·97-s + 10·100-s − 7·103-s + 17·109-s + ⋯
 L(s)  = 1 − 4-s − 1.94·13-s + 16-s − 1.60·19-s − 25-s − 1.25·31-s − 0.164·37-s + 0.762·43-s + 1.94·52-s + 1.79·61-s − 64-s + 1.34·67-s − 0.819·73-s + 1.60·76-s − 1.46·79-s + 1.42·97-s + 100-s − 0.689·103-s + 1.62·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$441$$    =    $$3^{2} \cdot 7^{2}$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{441} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 441,\ (\ :1/2),\ -1)$$

## 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$$
7 $$1$$
good2 $$1 + p T^{2}$$
5 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 + 7 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 7 T + p T^{2}$$
37 $$1 + T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 5 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 - 11 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 + 13 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 - 14 T + p 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

−10.44340961493860792194432079598, −9.751017393898932451994893756079, −8.947372931910945992290137358883, −7.991372095607934979439494362982, −7.07511528958958291144252655588, −5.72591573809169746727814745787, −4.78354615730792598375845969221, −3.87851392965924754220515773132, −2.26817487861975221274698654515, 0, 2.26817487861975221274698654515, 3.87851392965924754220515773132, 4.78354615730792598375845969221, 5.72591573809169746727814745787, 7.07511528958958291144252655588, 7.991372095607934979439494362982, 8.947372931910945992290137358883, 9.751017393898932451994893756079, 10.44340961493860792194432079598