# Properties

 Label 1-1400-1400.69-r1-0-0 Degree $1$ Conductor $1400$ Sign $0.535 + 0.844i$ Analytic cond. $150.450$ Root an. cond. $150.450$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.309 + 0.951i)3-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + (0.809 + 0.587i)41-s + 43-s + ⋯
 L(s)  = 1 + (−0.309 + 0.951i)3-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.809 − 0.587i)37-s + (−0.809 + 0.587i)39-s + (0.809 + 0.587i)41-s + 43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1400$$    =    $$2^{3} \cdot 5^{2} \cdot 7$$ Sign: $0.535 + 0.844i$ Analytic conductor: $$150.450$$ Root analytic conductor: $$150.450$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1400} (69, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1400,\ (1:\ ),\ 0.535 + 0.844i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.952347149 + 1.073311928i$$ $$L(\frac12)$$ $$\approx$$ $$1.952347149 + 1.073311928i$$ $$L(1)$$ $$\approx$$ $$1.061050533 + 0.3390015226i$$ $$L(1)$$ $$\approx$$ $$1.061050533 + 0.3390015226i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1$$
good3 $$1 + (-0.309 + 0.951i)T$$
11 $$1 + (0.809 - 0.587i)T$$
13 $$1 + (0.809 + 0.587i)T$$
17 $$1 + (0.309 + 0.951i)T$$
19 $$1 + (0.309 + 0.951i)T$$
23 $$1 + (0.809 - 0.587i)T$$
29 $$1 + (-0.309 + 0.951i)T$$
31 $$1 + (-0.309 - 0.951i)T$$
37 $$1 + (-0.809 - 0.587i)T$$
41 $$1 + (0.809 + 0.587i)T$$
43 $$1 + T$$
47 $$1 + (0.309 - 0.951i)T$$
53 $$1 + (0.309 - 0.951i)T$$
59 $$1 + (-0.809 - 0.587i)T$$
61 $$1 + (-0.809 + 0.587i)T$$
67 $$1 + (0.309 + 0.951i)T$$
71 $$1 + (0.309 - 0.951i)T$$
73 $$1 + (-0.809 + 0.587i)T$$
79 $$1 + (0.309 - 0.951i)T$$
83 $$1 + (-0.309 - 0.951i)T$$
89 $$1 + (0.809 - 0.587i)T$$
97 $$1 + (0.309 - 0.951i)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

−20.340326812882070473942973130640, −19.67001714194033850284181069436, −18.95104251244879571114460736257, −18.17031466579757621319558105986, −17.51518363139855900783596809916, −16.997426599128607714354041463957, −15.90980388666226697446058417448, −15.249142629956503100141187658441, −14.083185932445922359929006519616, −13.6797188059426338168717661711, −12.749005328449855760703996836, −12.111499001847088224604662871436, −11.31646561680312561666067969272, −10.69660314190389442402614349059, −9.40987389619272907706365139404, −8.84705533256251923859527509581, −7.67552296630533266225568804062, −7.18577885528418384110586558882, −6.32696146027637380491761654368, −5.50074593978163794974947585570, −4.643463826816439554399479691816, −3.380367376750758703136122567698, −2.51674172804563158190065687158, −1.35531216109057041260759396722, −0.695737256655293712972459440083, 0.7231622801997040356808576971, 1.80727784560861708935179370348, 3.32175082162978366225756456814, 3.78774017489333712253903534213, 4.65465571314489837624926493340, 5.83321647094291763049860475252, 6.154943704622629432815459852479, 7.34667725355032576358210958167, 8.624603039404778503585587280150, 8.9450929818184394540214594928, 9.94246509493576664010441872381, 10.75947828805078395747255621584, 11.318286458217387553940395647093, 12.13882797949781786506229893948, 13.033142054109225852727189753722, 14.2192462513067941085019196623, 14.55249918796163017439274706413, 15.47454345072881703865534753845, 16.46402556018242835058753008235, 16.63043320128745782566172280310, 17.533950655557402588026613221966, 18.52641958134875852018434642385, 19.21059876026107898776501676040, 20.1142806217439828585442596957, 20.929115306755887945460103699743