# Properties

 Label 2-435-435.434-c2-0-83 Degree $2$ Conductor $435$ Sign $1$ Analytic cond. $11.8528$ Root an. cond. $3.44280$ Motivic weight $2$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3·3-s + 4·4-s + 5·5-s + 9·9-s − 7·11-s + 12·12-s + 15·15-s + 16·16-s + 20·20-s − 41·23-s + 25·25-s + 27·27-s − 29·29-s − 21·33-s + 36·36-s − 71·37-s + 53·41-s − 59·43-s − 28·44-s + 45·45-s + 48·48-s + 49·49-s + 19·53-s − 35·55-s + 60·60-s + 64·64-s − 123·69-s + ⋯
 L(s)  = 1 + 3-s + 4-s + 5-s + 9-s − 0.636·11-s + 12-s + 15-s + 16-s + 20-s − 1.78·23-s + 25-s + 27-s − 29-s − 0.636·33-s + 36-s − 1.91·37-s + 1.29·41-s − 1.37·43-s − 0.636·44-s + 45-s + 48-s + 49-s + 0.358·53-s − 0.636·55-s + 60-s + 64-s − 1.78·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$435$$    =    $$3 \cdot 5 \cdot 29$$ Sign: $1$ Analytic conductor: $$11.8528$$ Root analytic conductor: $$3.44280$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: $\chi_{435} (434, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 435,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.468284061$$ $$L(\frac12)$$ $$\approx$$ $$3.468284061$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - p T$$
5 $$1 - p T$$
29 $$1 + p T$$
good2 $$( 1 - p T )( 1 + p T )$$
7 $$( 1 - p T )( 1 + p T )$$
11 $$1 + 7 T + p^{2} T^{2}$$
13 $$( 1 - p T )( 1 + p T )$$
17 $$( 1 - p T )( 1 + p T )$$
19 $$( 1 - p T )( 1 + p T )$$
23 $$1 + 41 T + p^{2} T^{2}$$
31 $$( 1 - p T )( 1 + p T )$$
37 $$1 + 71 T + p^{2} T^{2}$$
41 $$1 - 53 T + p^{2} T^{2}$$
43 $$1 + 59 T + p^{2} T^{2}$$
47 $$( 1 - p T )( 1 + p T )$$
53 $$1 - 19 T + p^{2} T^{2}$$
59 $$( 1 - p T )( 1 + p T )$$
61 $$( 1 - p T )( 1 + p T )$$
67 $$( 1 - p T )( 1 + p T )$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$1 - T + p^{2} T^{2}$$
79 $$( 1 - p T )( 1 + p T )$$
83 $$1 - 79 T + p^{2} T^{2}$$
89 $$1 - 62 T + p^{2} T^{2}$$
97 $$1 - 49 T + p^{2} 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

−10.57865229657446330904051304559, −10.14539942816032697080448737232, −9.165351489025971776773748132936, −8.126541929220751302112499800535, −7.29950780219645638875971702652, −6.32293368950194777674449095818, −5.31370734037183740441816077529, −3.68890166970416264627864850634, −2.48998736586266517105220072732, −1.74122832217435021953971945172, 1.74122832217435021953971945172, 2.48998736586266517105220072732, 3.68890166970416264627864850634, 5.31370734037183740441816077529, 6.32293368950194777674449095818, 7.29950780219645638875971702652, 8.126541929220751302112499800535, 9.165351489025971776773748132936, 10.14539942816032697080448737232, 10.57865229657446330904051304559