# Properties

 Degree $2$ Conductor $4650$ Sign $0.447 - 0.894i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## 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 − 6i·13-s + 16-s + 2i·17-s − i·18-s − 4·19-s − 4i·22-s + 8i·23-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 − 1.66i·13-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 0.917·19-s − 0.852i·22-s + 1.66i·23-s − 0.204·24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{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: $$4650$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 31$$ Sign: $0.447 - 0.894i$ Motivic weight: $$1$$ Character: $\chi_{4650} (3349, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4650,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.112846530$$ $$L(\frac12)$$ $$\approx$$ $$1.112846530$$ $$L(\frac{3}{2})$$ 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$$
31 $$1 + T$$
good7 $$1 - 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 8iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 - 10T + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 - 10iT - 53T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 2iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 4iT - 83T^{2}$$
89 $$1 - 14T + 89T^{2}$$
97 $$1 - 18iT - 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

−8.131172091433443907611229110211, −7.64225774108516106770381217401, −7.28985638622406679814529895354, −6.03265036783062671656038008801, −5.73308360394574246755242445904, −5.05503011365676459693430618962, −3.93407613333581363368019575718, −3.06084798008186798510762988705, −2.10672157202498530092111965838, −0.75200811348567704805352024882, 0.42491837607475486422221526359, 2.10071159374473486521558737460, 2.51835469249194975979516482251, 3.72116992769178786307264091475, 4.36700328846565953683990333043, 4.97036767889182386816932168076, 5.84179288751046260251331246618, 6.73066408545719865337403776170, 7.52881854441892907560908322998, 8.525647481673992696530012265165