# Properties

 Degree $2$ Conductor $2550$ Sign $0.894 + 0.447i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + i·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s − 4·11-s + i·12-s + 2i·13-s + 16-s + i·17-s − i·18-s − 4·19-s − 4i·22-s − 24-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 0.554i·13-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s − 0.917·19-s − 0.852i·22-s − 0.204·24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2550$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 17$$ Sign: $0.894 + 0.447i$ Motivic weight: $$1$$ Character: $\chi_{2550} (2449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2550,\ (\ :1/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.256666194$$ $$L(\frac12)$$ $$\approx$$ $$1.256666194$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 + iT$$
5 $$1$$
17 $$1 - iT$$
good7 $$1 - 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 - 2iT - 13T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 10T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 - 10T + 41T^{2}$$
43 $$1 + 12iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 + 12iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 10iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 + 4iT - 83T^{2}$$
89 $$1 - 6T + 89T^{2}$$
97 $$1 + 14iT - 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

−8.594596012387730146208750059529, −8.013495608938935115657463212527, −7.35693212743132773923222440700, −6.48349120520564199826544232607, −5.99743994823797004689189335285, −4.96271772592126370217239940426, −4.30483768612326852373799885486, −3.00107749095560770941295475832, −2.05669617851703378174457535933, −0.52462297082912301204589856869, 0.940197814882960083574972967081, 2.63209985225203183245732193592, 2.90052049639930777569910855694, 4.30393605622137117591546751804, 4.72689083643345171935497120211, 5.69750401876247945288874825377, 6.48525353000009579257296300101, 7.80323405699484606229033304585, 8.217657119251712969813260519890, 9.087910543834248305861046634847