# Properties

 Label 2-3200-40.29-c1-0-30 Degree $2$ Conductor $3200$ Sign $0.447 + 0.894i$ Analytic cond. $25.5521$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2.23·3-s − 2.82i·7-s + 2.00·9-s + 2.23i·11-s − 6.32·13-s + 5i·17-s + 2.23i·19-s + 6.32i·21-s − 5.65i·23-s + 2.23·27-s + 6.32i·29-s − 5.00i·33-s + 6.32·37-s + 14.1·39-s + 3·41-s + ⋯
 L(s)  = 1 − 1.29·3-s − 1.06i·7-s + 0.666·9-s + 0.674i·11-s − 1.75·13-s + 1.21i·17-s + 0.512i·19-s + 1.38i·21-s − 1.17i·23-s + 0.430·27-s + 1.17i·29-s − 0.870i·33-s + 1.03·37-s + 2.26·39-s + 0.468·41-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $0.447 + 0.894i$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6362760192$$ $$L(\frac12)$$ $$\approx$$ $$0.6362760192$$ $$L(\frac{3}{2})$$ 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 + 2.23T + 3T^{2}$$
7 $$1 + 2.82iT - 7T^{2}$$
11 $$1 - 2.23iT - 11T^{2}$$
13 $$1 + 6.32T + 13T^{2}$$
17 $$1 - 5iT - 17T^{2}$$
19 $$1 - 2.23iT - 19T^{2}$$
23 $$1 + 5.65iT - 23T^{2}$$
29 $$1 - 6.32iT - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 6.32T + 37T^{2}$$
41 $$1 - 3T + 41T^{2}$$
43 $$1 - 8.94T + 43T^{2}$$
47 $$1 + 2.82iT - 47T^{2}$$
53 $$1 + 12.6T + 53T^{2}$$
59 $$1 + 8.94iT - 59T^{2}$$
61 $$1 - 6.32iT - 61T^{2}$$
67 $$1 + 11.1T + 67T^{2}$$
71 $$1 + 14.1T + 71T^{2}$$
73 $$1 - 15iT - 73T^{2}$$
79 $$1 - 14.1T + 79T^{2}$$
83 $$1 + 6.70T + 83T^{2}$$
89 $$1 + T + 89T^{2}$$
97 $$1 + 10iT - 97T^{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

−8.424388070197217604442950781230, −7.49919806320799352825259882459, −7.04716087456950582745593103219, −6.26052540772479174785004494437, −5.52005721384635720018472587283, −4.54821711831870523265548632201, −4.29264343856643013661026065421, −2.85842460885069053354522728851, −1.60966168125923872046140163629, −0.36406028847228352297394652487, 0.70516200040660348069073472688, 2.33440120821207507857513306277, 3.00894460282619592593678535657, 4.56623238752684872723638007069, 5.02742749764096663069953405896, 5.83704132985375205997116213277, 6.21774970982900809527807989284, 7.36404333651414282818571525407, 7.77526052302832037500498520258, 9.164682270507401831741271956473