# Properties

 Label 2-432-1.1-c1-0-2 Degree $2$ Conductor $432$ Sign $1$ Analytic cond. $3.44953$ Root an. cond. $1.85729$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 7-s + 5·13-s + 7·19-s − 5·25-s + 4·31-s + 11·37-s − 8·43-s − 6·49-s − 61-s − 5·67-s − 7·73-s − 17·79-s + 5·91-s − 19·97-s + 13·103-s + 2·109-s + ⋯
 L(s)  = 1 + 0.377·7-s + 1.38·13-s + 1.60·19-s − 25-s + 0.718·31-s + 1.80·37-s − 1.21·43-s − 6/7·49-s − 0.128·61-s − 0.610·67-s − 0.819·73-s − 1.91·79-s + 0.524·91-s − 1.92·97-s + 1.28·103-s + 0.191·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$432$$    =    $$2^{4} \cdot 3^{3}$$ Sign: $1$ Analytic conductor: $$3.44953$$ Root analytic conductor: $$1.85729$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{432} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 432,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.529954037$$ $$L(\frac12)$$ $$\approx$$ $$1.529954037$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + p T^{2}$$
7 $$1 - T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 5 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 - 4 T + p T^{2}$$
37 $$1 - 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 + 5 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 + 17 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 19 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

−11.38842166918163472638466197964, −10.22208553622788501090350672882, −9.387437094399487619618753359356, −8.341275212548067016845372469550, −7.60583694658959035837478078570, −6.37318226789872439689553131647, −5.48198608682619476325367860976, −4.24977516365535388494352784618, −3.07984083787626508123182523604, −1.36018793518762280095521620239, 1.36018793518762280095521620239, 3.07984083787626508123182523604, 4.24977516365535388494352784618, 5.48198608682619476325367860976, 6.37318226789872439689553131647, 7.60583694658959035837478078570, 8.341275212548067016845372469550, 9.387437094399487619618753359356, 10.22208553622788501090350672882, 11.38842166918163472638466197964