# Properties

 Label 2-825-5.4-c3-0-69 Degree $2$ Conductor $825$ Sign $0.894 + 0.447i$ Analytic cond. $48.6765$ Root an. cond. $6.97686$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 3.59i·2-s + 3i·3-s − 4.89·4-s − 10.7·6-s − 16.1i·7-s + 11.1i·8-s − 9·9-s + 11·11-s − 14.6i·12-s + 54.1i·13-s + 57.9·14-s − 79.2·16-s − 107. i·17-s − 32.3i·18-s − 48.7·19-s + ⋯
 L(s)  = 1 + 1.26i·2-s + 0.577i·3-s − 0.611·4-s − 0.732·6-s − 0.871i·7-s + 0.493i·8-s − 0.333·9-s + 0.301·11-s − 0.353i·12-s + 1.15i·13-s + 1.10·14-s − 1.23·16-s − 1.52i·17-s − 0.423i·18-s − 0.588·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$825$$    =    $$3 \cdot 5^{2} \cdot 11$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$48.6765$$ Root analytic conductor: $$6.97686$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{825} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 825,\ (\ :3/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.5719441944$$ $$L(\frac12)$$ $$\approx$$ $$0.5719441944$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3iT$$
5 $$1$$
11 $$1 - 11T$$
good2 $$1 - 3.59iT - 8T^{2}$$
7 $$1 + 16.1iT - 343T^{2}$$
13 $$1 - 54.1iT - 2.19e3T^{2}$$
17 $$1 + 107. iT - 4.91e3T^{2}$$
19 $$1 + 48.7T + 6.85e3T^{2}$$
23 $$1 + 11.9iT - 1.21e4T^{2}$$
29 $$1 + 239.T + 2.43e4T^{2}$$
31 $$1 + 82.0T + 2.97e4T^{2}$$
37 $$1 + 21.7iT - 5.06e4T^{2}$$
41 $$1 + 124.T + 6.89e4T^{2}$$
43 $$1 + 224. iT - 7.95e4T^{2}$$
47 $$1 + 186. iT - 1.03e5T^{2}$$
53 $$1 + 233. iT - 1.48e5T^{2}$$
59 $$1 + 232.T + 2.05e5T^{2}$$
61 $$1 - 163.T + 2.26e5T^{2}$$
67 $$1 + 876. iT - 3.00e5T^{2}$$
71 $$1 + 733.T + 3.57e5T^{2}$$
73 $$1 + 1.16e3iT - 3.89e5T^{2}$$
79 $$1 - 588.T + 4.93e5T^{2}$$
83 $$1 - 1.16e3iT - 5.71e5T^{2}$$
89 $$1 - 1.04e3T + 7.04e5T^{2}$$
97 $$1 - 1.54e3iT - 9.12e5T^{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

−9.390462989063643004273335217965, −8.997267998056960619072919001887, −7.82579541193419336610784081188, −7.10059928253472867089714288081, −6.49870885328701099282406900903, −5.37169642272928413051914471634, −4.58685377708898067915749473034, −3.66750897630696223020547677680, −2.07141164971225816415223993932, −0.14572755111208684153193827265, 1.30572348185022477284402139914, 2.17489588850634757249021188060, 3.14235106942633049239877996582, 4.09913153835463025329366251626, 5.56987505203794453990543621571, 6.26520819058045727158259345725, 7.43406930515446371120130445826, 8.415960758764556925244101400029, 9.144086565267330360712416621209, 10.13605499264659932894403024429