# Properties

 Label 2-3200-200.173-c0-0-0 Degree $2$ Conductor $3200$ Sign $0.684 + 0.728i$ 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 + (0.951 − 0.309i)5-s + (0.587 − 0.809i)9-s + (0.896 − 0.142i)13-s + (−1.76 − 0.896i)17-s + (0.809 − 0.587i)25-s + (0.5 − 1.53i)29-s + (0.309 + 1.95i)37-s + (−1.53 − 1.11i)41-s + (0.309 − 0.951i)45-s + i·49-s + (0.809 + 1.58i)53-s + (−0.363 − 0.5i)61-s + (0.809 − 0.412i)65-s + (−0.896 − 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯
 L(s)  = 1 + (0.951 − 0.309i)5-s + (0.587 − 0.809i)9-s + (0.896 − 0.142i)13-s + (−1.76 − 0.896i)17-s + (0.809 − 0.587i)25-s + (0.5 − 1.53i)29-s + (0.309 + 1.95i)37-s + (−1.53 − 1.11i)41-s + (0.309 − 0.951i)45-s + i·49-s + (0.809 + 1.58i)53-s + (−0.363 − 0.5i)61-s + (0.809 − 0.412i)65-s + (−0.896 − 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $0.684 + 0.728i$ Analytic conductor: $$1.59700$$ Root analytic conductor: $$1.26372$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1473, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :0),\ 0.684 + 0.728i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.542433431$$ $$L(\frac12)$$ $$\approx$$ $$1.542433431$$ $$L(1)$$ 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 + (-0.951 + 0.309i)T$$
good3 $$1 + (-0.587 + 0.809i)T^{2}$$
7 $$1 - iT^{2}$$
11 $$1 + (-0.309 + 0.951i)T^{2}$$
13 $$1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2}$$
17 $$1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2}$$
19 $$1 + (-0.809 + 0.587i)T^{2}$$
23 $$1 + (0.951 + 0.309i)T^{2}$$
29 $$1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}$$
31 $$1 + (-0.809 + 0.587i)T^{2}$$
37 $$1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2}$$
41 $$1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 + (-0.587 + 0.809i)T^{2}$$
53 $$1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2}$$
59 $$1 + (0.309 + 0.951i)T^{2}$$
61 $$1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2}$$
67 $$1 + (-0.587 - 0.809i)T^{2}$$
71 $$1 + (-0.809 - 0.587i)T^{2}$$
73 $$1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2}$$
79 $$1 + (0.809 + 0.587i)T^{2}$$
83 $$1 + (0.587 + 0.809i)T^{2}$$
89 $$1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2}$$
97 $$1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)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

−8.908909039145596992389405141287, −8.175052354192274336983341805327, −7.02921019544704755726196527546, −6.45162545666089834369227336558, −5.90666254266253228148474384820, −4.79813026587707380752505612964, −4.23757910156487224304790595559, −3.06768138586088866772665989695, −2.10219786304855690986400064380, −1.00176787655974913117146060524, 1.60842696854187760692360529991, 2.18147198244407397291609807311, 3.38283107348994227472426037544, 4.35772636762039641551769448510, 5.14552965277944324194331920234, 5.99592970667031329937516755013, 6.72075262256837512559005734315, 7.23112017668530620485462132281, 8.520961612356084321238471350587, 8.727146995308200472057641120842