# Properties

 Label 1-1045-1045.217-r1-0-0 Degree $1$ Conductor $1045$ Sign $-0.993 + 0.116i$ Analytic cond. $112.300$ Root an. cond. $112.300$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s − i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + ⋯
 L(s)  = 1 + (−0.207 − 0.978i)2-s + (−0.406 + 0.913i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)6-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)9-s − i·12-s + (0.743 − 0.669i)13-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.743 − 0.669i)17-s + (−0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (−0.978 + 0.207i)24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $-0.993 + 0.116i$ Analytic conductor: $$112.300$$ Root analytic conductor: $$112.300$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (217, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1045,\ (1:\ ),\ -0.993 + 0.116i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.01137036170 + 0.1945469845i$$ $$L(\frac12)$$ $$\approx$$ $$0.01137036170 + 0.1945469845i$$ $$L(1)$$ $$\approx$$ $$0.6420708924 + 0.01585429887i$$ $$L(1)$$ $$\approx$$ $$0.6420708924 + 0.01585429887i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
11 $$1$$
19 $$1$$
good2 $$1 + (-0.207 - 0.978i)T$$
3 $$1 + (-0.406 + 0.913i)T$$
7 $$1 + (-0.587 + 0.809i)T$$
13 $$1 + (0.743 - 0.669i)T$$
17 $$1 + (-0.743 - 0.669i)T$$
23 $$1 + (0.866 + 0.5i)T$$
29 $$1 + (-0.913 + 0.406i)T$$
31 $$1 + (-0.309 + 0.951i)T$$
37 $$1 + (-0.587 + 0.809i)T$$
41 $$1 + (0.913 + 0.406i)T$$
43 $$1 + (0.866 - 0.5i)T$$
47 $$1 + (0.994 - 0.104i)T$$
53 $$1 + (-0.743 + 0.669i)T$$
59 $$1 + (-0.104 + 0.994i)T$$
61 $$1 + (0.978 + 0.207i)T$$
67 $$1 + (0.866 + 0.5i)T$$
71 $$1 + (-0.669 + 0.743i)T$$
73 $$1 + (-0.994 - 0.104i)T$$
79 $$1 + (0.978 - 0.207i)T$$
83 $$1 + (-0.951 + 0.309i)T$$
89 $$1 + (-0.5 + 0.866i)T$$
97 $$1 + (0.207 + 0.978i)T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−20.94995865749387712399522286044, −19.8818352424409767918968601257, −19.04564413206656139980360946683, −18.71286351856274095064707066877, −17.57937163477713515484812474454, −17.165472877042489592538957388652, −16.396102759832572893421659727586, −15.72280254003409335042026416847, −14.55936915170188651847024519171, −13.875767312174188290338753959626, −13.06967525601588528661455364114, −12.71323134590134389899667272645, −11.20350240613975116816040217198, −10.689682945365772894004274963895, −9.454505073783418897548293165547, −8.70939932550539724820277659257, −7.723130258661130928838467439948, −7.05891844617093385800542496134, −6.35543220505180801040790791205, −5.74628039083353230816298897627, −4.51185897816046415339378124959, −3.684307140638907370653692419378, −2.05968808377089151668886070023, −0.9175988851241992621404300675, −0.06312849245190422618629520105, 1.12471666875595628166851451969, 2.61496028182618556524172710555, 3.25502956805343191980914502085, 4.147174203461774644632154281613, 5.20658255931803171234718163800, 5.78266080576474949943702951887, 7.089887731560934775045400776561, 8.53491698733248300857909815759, 9.05493970223664548661141944874, 9.7123216242913402544111288283, 10.69228940727065347413875929117, 11.18201693178808931746962489661, 12.057878554028242356059972897978, 12.81075464143029821034367656768, 13.62647885612795014633574860582, 14.7218041778915166103967517440, 15.60015625828959215610528132377, 16.189895100983688832405240361502, 17.21711344922002786378404494753, 17.90221847450269199782309472436, 18.627923719191057645639864471814, 19.482588129528805675567083401201, 20.39719257712356579755138260342, 20.846961848727135163031184236235, 21.78724944327482356505376510561