# Properties

 Label 1-2205-2205.2084-r0-0-0 Degree $1$ Conductor $2205$ Sign $0.965 - 0.260i$ Analytic cond. $10.2399$ Root an. cond. $10.2399$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)22-s + (0.955 + 0.294i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.222 + 0.974i)32-s + (0.988 − 0.149i)34-s + ⋯
 L(s)  = 1 + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)8-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.900 + 0.433i)16-s + (0.733 + 0.680i)17-s + (0.5 + 0.866i)19-s + (0.733 − 0.680i)22-s + (0.955 + 0.294i)23-s + (−0.733 + 0.680i)26-s + (0.733 + 0.680i)29-s − 31-s + (−0.222 + 0.974i)32-s + (0.988 − 0.149i)34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$2205$$    =    $$3^{2} \cdot 5 \cdot 7^{2}$$ Sign: $0.965 - 0.260i$ Analytic conductor: $$10.2399$$ Root analytic conductor: $$10.2399$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2205} (2084, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 2205,\ (0:\ ),\ 0.965 - 0.260i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.042730296 - 0.2707812889i$$ $$L(\frac12)$$ $$\approx$$ $$2.042730296 - 0.2707812889i$$ $$L(1)$$ $$\approx$$ $$1.331820785 - 0.4655950090i$$ $$L(1)$$ $$\approx$$ $$1.331820785 - 0.4655950090i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1$$
good2 $$1 + (0.623 - 0.781i)T$$
11 $$1 + (0.988 + 0.149i)T$$
13 $$1 + (-0.988 - 0.149i)T$$
17 $$1 + (0.733 + 0.680i)T$$
19 $$1 + (0.5 + 0.866i)T$$
23 $$1 + (0.955 + 0.294i)T$$
29 $$1 + (0.733 + 0.680i)T$$
31 $$1 - T$$
37 $$1 + (-0.955 + 0.294i)T$$
41 $$1 + (0.0747 + 0.997i)T$$
43 $$1 + (-0.0747 + 0.997i)T$$
47 $$1 + (-0.623 + 0.781i)T$$
53 $$1 + (0.955 + 0.294i)T$$
59 $$1 + (-0.900 + 0.433i)T$$
61 $$1 + (0.222 - 0.974i)T$$
67 $$1 - T$$
71 $$1 + (0.222 + 0.974i)T$$
73 $$1 + (-0.988 + 0.149i)T$$
79 $$1 + T$$
83 $$1 + (0.988 - 0.149i)T$$
89 $$1 + (0.365 - 0.930i)T$$
97 $$1 + (-0.5 + 0.866i)T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−19.73607658830725398560107111964, −19.08465397506814086821543184209, −18.08478540573805453592222723966, −17.40654459224075920392691944433, −16.76272611260869799198794633843, −16.234433643262526167610481980361, −15.24801628971252327686890474350, −14.77326301298793995818584870316, −13.95126736837089866543219964634, −13.52488627888387070382577772737, −12.376150199065804944676224731208, −12.01576831822262468890082104756, −11.22986727436432297742902287707, −10.07029496412087776006918020243, −9.12586243922961240666101996839, −8.72962852796413673054797468017, −7.41743766374683270710247915555, −7.16964960686959377850440470045, −6.29588257709761172085423267190, −5.286583148167460519821410798451, −4.82964614799852454196751265102, −3.80561029177547324488494969103, −3.079182569046282266330476631062, −2.119826314574011747686816392418, −0.59756089090680855039595857883, 1.15566514883914779883085532824, 1.7246440465543992975176026400, 2.97182183955731210662834424614, 3.5032646946170365308472760007, 4.47842364654545628606960671799, 5.18643026568768731293462274104, 6.00472803514240133795169900839, 6.83044781919610876044636307607, 7.739432008531489999605750707060, 8.84917123907447483891265874451, 9.57967735526486086676900109584, 10.18470992142309662594650409638, 10.98458374451672559222846346638, 11.85026470702293929786488368935, 12.34092952958153497315695876480, 12.974680763200105118878511286752, 13.94345295352648952968009552717, 14.70954442603646186201147827228, 14.843192323019017413362324384, 16.06974956386806670283539499235, 16.8564646393085838205343653735, 17.620415112204112594872882411795, 18.462552933519545235785623462400, 19.25876619632612657970302267621, 19.71072938111106842341722014444