# Properties

 Label 2-200-1.1-c3-0-0 Degree $2$ Conductor $200$ Sign $1$ Analytic cond. $11.8003$ Root an. cond. $3.43516$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 9·3-s − 26·7-s + 54·9-s − 59·11-s − 28·13-s − 5·17-s + 109·19-s + 234·21-s + 194·23-s − 243·27-s − 32·29-s + 10·31-s + 531·33-s + 198·37-s + 252·39-s + 117·41-s − 388·43-s + 68·47-s + 333·49-s + 45·51-s + 18·53-s − 981·57-s + 392·59-s − 710·61-s − 1.40e3·63-s + 253·67-s − 1.74e3·69-s + ⋯
 L(s)  = 1 − 1.73·3-s − 1.40·7-s + 2·9-s − 1.61·11-s − 0.597·13-s − 0.0713·17-s + 1.31·19-s + 2.43·21-s + 1.75·23-s − 1.73·27-s − 0.204·29-s + 0.0579·31-s + 2.80·33-s + 0.879·37-s + 1.03·39-s + 0.445·41-s − 1.37·43-s + 0.211·47-s + 0.970·49-s + 0.123·51-s + 0.0466·53-s − 2.27·57-s + 0.864·59-s − 1.49·61-s − 2.80·63-s + 0.461·67-s − 3.04·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$200$$    =    $$2^{3} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$11.8003$$ Root analytic conductor: $$3.43516$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 200,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.5016799116$$ $$L(\frac12)$$ $$\approx$$ $$0.5016799116$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + p^{2} T + p^{3} T^{2}$$
7 $$1 + 26 T + p^{3} T^{2}$$
11 $$1 + 59 T + p^{3} T^{2}$$
13 $$1 + 28 T + p^{3} T^{2}$$
17 $$1 + 5 T + p^{3} T^{2}$$
19 $$1 - 109 T + p^{3} T^{2}$$
23 $$1 - 194 T + p^{3} T^{2}$$
29 $$1 + 32 T + p^{3} T^{2}$$
31 $$1 - 10 T + p^{3} T^{2}$$
37 $$1 - 198 T + p^{3} T^{2}$$
41 $$1 - 117 T + p^{3} T^{2}$$
43 $$1 + 388 T + p^{3} T^{2}$$
47 $$1 - 68 T + p^{3} T^{2}$$
53 $$1 - 18 T + p^{3} T^{2}$$
59 $$1 - 392 T + p^{3} T^{2}$$
61 $$1 + 710 T + p^{3} T^{2}$$
67 $$1 - 253 T + p^{3} T^{2}$$
71 $$1 + 612 T + p^{3} T^{2}$$
73 $$1 - 549 T + p^{3} T^{2}$$
79 $$1 - 414 T + p^{3} T^{2}$$
83 $$1 - 121 T + p^{3} T^{2}$$
89 $$1 + 81 T + p^{3} T^{2}$$
97 $$1 - 1502 T + p^{3} 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

−12.00586198549451905557587564057, −11.01575675197227094063879906510, −10.21560922558128184478271897619, −9.453388275223862237477712941719, −7.53970899492424876315488848537, −6.70084975664286266857129467903, −5.61071272134961031897890661096, −4.88983074140591676150401453906, −3.01832712898436031290790617307, −0.56457555693591333235768816055, 0.56457555693591333235768816055, 3.01832712898436031290790617307, 4.88983074140591676150401453906, 5.61071272134961031897890661096, 6.70084975664286266857129467903, 7.53970899492424876315488848537, 9.453388275223862237477712941719, 10.21560922558128184478271897619, 11.01575675197227094063879906510, 12.00586198549451905557587564057