# Properties

 Label 2-3200-40.19-c0-0-3 Degree $2$ Conductor $3200$ Sign $0.894 + 0.447i$ Analytic cond. $1.59700$ Root an. cond. $1.26372$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 9-s − 2i·17-s + 2·41-s − 49-s + 2i·73-s + 81-s + 2·89-s − 2i·97-s − 2i·113-s + ⋯
 L(s)  = 1 + 9-s − 2i·17-s + 2·41-s − 49-s + 2i·73-s + 81-s + 2·89-s − 2i·97-s − 2i·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$1.59700$$ Root analytic conductor: $$1.26372$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1599, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :0),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.355693605$$ $$L(\frac12)$$ $$\approx$$ $$1.355693605$$ $$L(1)$$ 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 - T^{2}$$
7 $$1 + T^{2}$$
11 $$1 + T^{2}$$
13 $$1 + T^{2}$$
17 $$1 + 2iT - T^{2}$$
19 $$1 + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 - T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 - 2T + T^{2}$$
43 $$1 - T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - 2iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - 2T + T^{2}$$
97 $$1 + 2iT - 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

−8.926511194662820850240342560570, −7.889187846629726115227327439245, −7.29277222425626572370166565457, −6.71234694631139583741292751851, −5.72451586503508638787691703991, −4.85778357732164184091919133485, −4.24337099456036465231196824934, −3.16670819682663778767983452595, −2.25055935417778432185130372300, −0.966296917955607138933026412603, 1.30601088220893757435696801895, 2.22811695954816862519000804159, 3.53556836780382205191151687096, 4.16692752720597607692152074251, 4.99387653059748385391181737894, 6.07439773450342772959049841622, 6.51204956738730704559948272503, 7.58264317696518773638369638542, 7.999787932225622694317180560457, 8.963496949292052677763011377799