# Properties

 Label 1-1045-1045.94-r0-0-0 Degree $1$ Conductor $1045$ Sign $0.0219 - 0.999i$ Analytic cond. $4.85295$ Root an. cond. $4.85295$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 12-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + 21-s − 23-s + (0.809 − 0.587i)24-s + ⋯
 L(s)  = 1 + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 12-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + 21-s − 23-s + (0.809 − 0.587i)24-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1045$$    =    $$5 \cdot 11 \cdot 19$$ Sign: $0.0219 - 0.999i$ Analytic conductor: $$4.85295$$ Root analytic conductor: $$4.85295$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1045} (94, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1045,\ (0:\ ),\ 0.0219 - 0.999i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.758066028 - 1.719832472i$$ $$L(\frac12)$$ $$\approx$$ $$1.758066028 - 1.719832472i$$ $$L(1)$$ $$\approx$$ $$1.629514152 - 0.6145153309i$$ $$L(1)$$ $$\approx$$ $$1.629514152 - 0.6145153309i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
11 $$1$$
19 $$1$$
good2 $$1 + (0.809 - 0.587i)T$$
3 $$1 + (0.309 + 0.951i)T$$
7 $$1 + (0.309 - 0.951i)T$$
13 $$1 + (0.809 - 0.587i)T$$
17 $$1 + (-0.809 - 0.587i)T$$
23 $$1 - T$$
29 $$1 + (0.309 - 0.951i)T$$
31 $$1 + (0.809 - 0.587i)T$$
37 $$1 + (0.309 - 0.951i)T$$
41 $$1 + (0.309 + 0.951i)T$$
43 $$1 + T$$
47 $$1 + (-0.309 - 0.951i)T$$
53 $$1 + (-0.809 + 0.587i)T$$
59 $$1 + (-0.309 + 0.951i)T$$
61 $$1 + (0.809 + 0.587i)T$$
67 $$1 + T$$
71 $$1 + (0.809 + 0.587i)T$$
73 $$1 + (0.309 - 0.951i)T$$
79 $$1 + (-0.809 + 0.587i)T$$
83 $$1 + (-0.809 - 0.587i)T$$
89 $$1 - T$$
97 $$1 + (-0.809 + 0.587i)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

−21.86523547496070693570958213708, −21.05425690868370574967548733104, −20.31245660336784542343775679913, −19.37116469871119597733434840862, −18.496539520906940102647418513018, −17.80660288004752909779486944980, −17.16492908340243969426842308586, −15.90058213982295355323801375917, −15.51219645817455272256740864449, −14.27765387253752607822692027724, −14.129555613207375350235554753951, −12.9855306423227929744503396662, −12.48879956758018076913166487669, −11.667678791477422776129531541238, −11.00578554845193018925803503658, −9.2858002345733657618216001133, −8.42967739410292905320145081636, −8.07250123276875296713046323226, −6.79544889630490647601635609885, −6.32011558021839988908850446920, −5.50760207971898920559518940119, −4.41747545748980578920344753649, −3.36674553449239030678131716305, −2.401738086255448559506162587656, −1.61036198836225163651328647215, 0.73017361818717432113446046031, 2.15453409042674043502035412178, 3.0604123167326531945428755504, 4.11811204978946636973566824357, 4.37821695324002447874244229085, 5.52206545044516520456675116095, 6.33121846304769382243790523795, 7.584658580766517080379044654680, 8.542804725397515278744260677932, 9.684095477996386263350512564723, 10.20803529648277056158540909510, 11.109521943891513868498696051318, 11.476757777844845053238266620101, 12.79280737305743668250719559967, 13.72941288638916370910754072371, 13.992004642671172835342167819678, 15.02810981313017565889037045621, 15.70646525710010127992495083957, 16.333515813303329204140793321686, 17.44135030534633101442622863299, 18.29911348820409810824319538820, 19.53244802751086340491574025390, 20.01962463885655597000524105671, 20.700610911737716766348731074718, 21.183647824923817033779170473126