# Properties

 Label 2-2200-1.1-c3-0-95 Degree $2$ Conductor $2200$ Sign $-1$ Analytic cond. $129.804$ Root an. cond. $11.3931$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 16·7-s − 27·9-s − 11·11-s + 70·13-s + 10·17-s − 12·19-s + 84·23-s + 30·29-s − 72·31-s − 310·37-s + 18·41-s + 388·43-s + 516·47-s − 87·49-s + 298·53-s + 204·59-s − 210·61-s + 432·63-s + 432·67-s − 440·71-s − 46·73-s + 176·77-s − 616·79-s + 729·81-s − 740·83-s − 6·89-s − 1.12e3·91-s + ⋯
 L(s)  = 1 − 0.863·7-s − 9-s − 0.301·11-s + 1.49·13-s + 0.142·17-s − 0.144·19-s + 0.761·23-s + 0.192·29-s − 0.417·31-s − 1.37·37-s + 0.0685·41-s + 1.37·43-s + 1.60·47-s − 0.253·49-s + 0.772·53-s + 0.450·59-s − 0.440·61-s + 0.863·63-s + 0.787·67-s − 0.735·71-s − 0.0737·73-s + 0.260·77-s − 0.877·79-s + 81-s − 0.978·83-s − 0.00714·89-s − 1.29·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2200$$    =    $$2^{3} \cdot 5^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$129.804$$ Root analytic conductor: $$11.3931$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2200,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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$$
11 $$1 + p T$$
good3 $$1 + p^{3} T^{2}$$
7 $$1 + 16 T + p^{3} T^{2}$$
13 $$1 - 70 T + p^{3} T^{2}$$
17 $$1 - 10 T + p^{3} T^{2}$$
19 $$1 + 12 T + p^{3} T^{2}$$
23 $$1 - 84 T + p^{3} T^{2}$$
29 $$1 - 30 T + p^{3} T^{2}$$
31 $$1 + 72 T + p^{3} T^{2}$$
37 $$1 + 310 T + p^{3} T^{2}$$
41 $$1 - 18 T + p^{3} T^{2}$$
43 $$1 - 388 T + p^{3} T^{2}$$
47 $$1 - 516 T + p^{3} T^{2}$$
53 $$1 - 298 T + p^{3} T^{2}$$
59 $$1 - 204 T + p^{3} T^{2}$$
61 $$1 + 210 T + p^{3} T^{2}$$
67 $$1 - 432 T + p^{3} T^{2}$$
71 $$1 + 440 T + p^{3} T^{2}$$
73 $$1 + 46 T + p^{3} T^{2}$$
79 $$1 + 616 T + p^{3} T^{2}$$
83 $$1 + 740 T + p^{3} T^{2}$$
89 $$1 + 6 T + p^{3} T^{2}$$
97 $$1 + 490 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

−8.626774150860195707272908019013, −7.51190524530924304978091960943, −6.67880055871588204224199725442, −5.90421545544309811136781554345, −5.37344578587189307145362735960, −4.06153373867016503505423095019, −3.30776209245288460664107103907, −2.51904405119199771990298319310, −1.12106020782597490521051181605, 0, 1.12106020782597490521051181605, 2.51904405119199771990298319310, 3.30776209245288460664107103907, 4.06153373867016503505423095019, 5.37344578587189307145362735960, 5.90421545544309811136781554345, 6.67880055871588204224199725442, 7.51190524530924304978091960943, 8.626774150860195707272908019013