# Properties

 Label 2-3200-200.53-c0-0-1 Degree $2$ Conductor $3200$ Sign $0.844 - 0.535i$ Analytic cond. $1.59700$ Root an. cond. $1.26372$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)9-s + (1.76 + 0.896i)13-s + (−0.278 + 1.76i)17-s + (−0.309 + 0.951i)25-s + (−0.5 − 0.363i)29-s + (0.809 − 1.58i)37-s + (0.363 + 1.11i)41-s + (0.809 + 0.587i)45-s + i·49-s + (0.309 − 0.0489i)53-s + (1.53 + 0.5i)61-s + (−0.309 − 1.95i)65-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯
 L(s)  = 1 + (−0.587 − 0.809i)5-s + (−0.951 + 0.309i)9-s + (1.76 + 0.896i)13-s + (−0.278 + 1.76i)17-s + (−0.309 + 0.951i)25-s + (−0.5 − 0.363i)29-s + (0.809 − 1.58i)37-s + (0.363 + 1.11i)41-s + (0.809 + 0.587i)45-s + i·49-s + (0.309 − 0.0489i)53-s + (1.53 + 0.5i)61-s + (−0.309 − 1.95i)65-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $0.844 - 0.535i$ Analytic conductor: $$1.59700$$ Root analytic conductor: $$1.26372$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (2753, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :0),\ 0.844 - 0.535i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.031809867$$ $$L(\frac12)$$ $$\approx$$ $$1.031809867$$ $$L(1)$$ 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 + (0.587 + 0.809i)T$$
good3 $$1 + (0.951 - 0.309i)T^{2}$$
7 $$1 - iT^{2}$$
11 $$1 + (0.809 + 0.587i)T^{2}$$
13 $$1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2}$$
17 $$1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2}$$
19 $$1 + (0.309 - 0.951i)T^{2}$$
23 $$1 + (0.587 - 0.809i)T^{2}$$
29 $$1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}$$
31 $$1 + (0.309 - 0.951i)T^{2}$$
37 $$1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2}$$
41 $$1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + (0.951 - 0.309i)T^{2}$$
53 $$1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2}$$
59 $$1 + (-0.809 + 0.587i)T^{2}$$
61 $$1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2}$$
67 $$1 + (0.951 + 0.309i)T^{2}$$
71 $$1 + (0.309 + 0.951i)T^{2}$$
73 $$1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2}$$
79 $$1 + (-0.309 - 0.951i)T^{2}$$
83 $$1 + (-0.951 - 0.309i)T^{2}$$
89 $$1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2}$$
97 $$1 + (0.896 - 0.142i)T + (0.951 - 0.309i)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

−8.914206415050773209144503505020, −8.167494162809066777289958127026, −7.76575590326420346921401202641, −6.40549396296497669456561903543, −6.00551719610959777270606485849, −5.10878437146811207599040779400, −3.96127964812492233976519614987, −3.77967290164660695972365977510, −2.26630454388658692102564065705, −1.19697567233760887196401608385, 0.72785290252308125464931330632, 2.47664765871470159988706684440, 3.24754200186197610132636050923, 3.81246579954198126158675274481, 5.04381964245982842898416246984, 5.82484557196755966205407976371, 6.56443041623214789759677076574, 7.23982243917906902942949120683, 8.167100797930879268692217980926, 8.586172837859195374764935973421