# Properties

 Label 2-320-5.4-c1-0-9 Degree $2$ Conductor $320$ Sign $-1$ Analytic cond. $2.55521$ Root an. cond. $1.59850$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3.23i·3-s − 2.23·5-s − 0.763i·7-s − 7.47·9-s + 7.23i·15-s − 2.47·21-s − 5.70i·23-s + 5.00·25-s + 14.4i·27-s − 6·29-s + 1.70i·35-s − 4.47·41-s − 11.2i·43-s + 16.7·45-s − 13.7i·47-s + ⋯
 L(s)  = 1 − 1.86i·3-s − 0.999·5-s − 0.288i·7-s − 2.49·9-s + 1.86i·15-s − 0.539·21-s − 1.19i·23-s + 1.00·25-s + 2.78i·27-s − 1.11·29-s + 0.288i·35-s − 0.698·41-s − 1.71i·43-s + 2.49·45-s − 1.99i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$320$$    =    $$2^{6} \cdot 5$$ Sign: $-1$ Analytic conductor: $$2.55521$$ Root analytic conductor: $$1.59850$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{320} (129, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 320,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.765918i$$ $$L(\frac12)$$ $$\approx$$ $$0.765918i$$ $$L(\frac{3}{2})$$ 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 + 2.23T$$
good3 $$1 + 3.23iT - 3T^{2}$$
7 $$1 + 0.763iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 + 5.70iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + 4.47T + 41T^{2}$$
43 $$1 + 11.2iT - 43T^{2}$$
47 $$1 + 13.7iT - 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 13.4T + 61T^{2}$$
67 $$1 + 8.18iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 17.7iT - 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 - 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

−11.52973018239200456695984912183, −10.55075035709947792383124374864, −8.799248625798489187961820827836, −8.145333263117380176796508711167, −7.22137510876829682312366512447, −6.70028884169748980067068633430, −5.37405656422761824414783060441, −3.68197503671921374509558661383, −2.21309292232618562980695288057, −0.54203674593918975402133761281, 3.07259212614449354672858705353, 3.96393813181188993381959928447, 4.86724775767359282115239252340, 5.88291129271952006489432174005, 7.55268921339908238736794427242, 8.595704134806719994888197460077, 9.375059098659536273399526608005, 10.20543804354717127961279603811, 11.28380129977894010022732637833, 11.54081180536531209042938298204