# Properties

 Label 1-1480-1480.237-r0-0-0 Degree $1$ Conductor $1480$ Sign $0.0702 - 0.997i$ Analytic cond. $6.87309$ Root an. cond. $6.87309$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (−0.939 + 0.342i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (0.984 − 0.173i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯
 L(s)  = 1 + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.173 + 0.984i)13-s + (0.173 − 0.984i)17-s + (−0.642 − 0.766i)19-s + (−0.939 + 0.342i)21-s + (0.5 + 0.866i)23-s + (0.866 − 0.5i)27-s + (−0.866 − 0.5i)29-s + i·31-s + (0.984 − 0.173i)33-s + (0.642 − 0.766i)39-s + (−0.173 − 0.984i)41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1480$$    =    $$2^{3} \cdot 5 \cdot 37$$ Sign: $0.0702 - 0.997i$ Analytic conductor: $$6.87309$$ Root analytic conductor: $$6.87309$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1480} (237, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 1480,\ (0:\ ),\ 0.0702 - 0.997i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7796665256 - 0.7266748244i$$ $$L(\frac12)$$ $$\approx$$ $$0.7796665256 - 0.7266748244i$$ $$L(1)$$ $$\approx$$ $$0.8170654701 - 0.2717536823i$$ $$L(1)$$ $$\approx$$ $$0.8170654701 - 0.2717536823i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
37 $$1$$
good3 $$1 + (-0.642 - 0.766i)T$$
7 $$1 + (0.342 - 0.939i)T$$
11 $$1 + (-0.5 + 0.866i)T$$
13 $$1 + (0.173 + 0.984i)T$$
17 $$1 + (0.173 - 0.984i)T$$
19 $$1 + (-0.642 - 0.766i)T$$
23 $$1 + (0.5 + 0.866i)T$$
29 $$1 + (-0.866 - 0.5i)T$$
31 $$1 + iT$$
41 $$1 + (-0.173 - 0.984i)T$$
43 $$1 + T$$
47 $$1 + (0.866 - 0.5i)T$$
53 $$1 + (-0.342 - 0.939i)T$$
59 $$1 + (0.342 + 0.939i)T$$
61 $$1 + (0.984 - 0.173i)T$$
67 $$1 + (0.342 - 0.939i)T$$
71 $$1 + (0.766 - 0.642i)T$$
73 $$1 + iT$$
79 $$1 + (0.342 - 0.939i)T$$
83 $$1 + (0.984 + 0.173i)T$$
89 $$1 + (0.342 + 0.939i)T$$
97 $$1 + (-0.5 - 0.866i)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

−20.82490388552817795660569248091, −20.495713211381891846161456437946, −19.08600857012231105456494120508, −18.61270629064514977441861542128, −17.778675516132530609649274434941, −17.003860684325262969060339448452, −16.35963347197677065321817363375, −15.50743306979064760902831283942, −14.984496935192367493892076795840, −14.32141240305635383504863540443, −12.868623275083661324427978473527, −12.58904044927010500073560864329, −11.47310779879384156466330195007, −10.851465011735674986644412809393, −10.291285530579027244095845518913, −9.25820773768089782666261712916, −8.4764795052110410743438427725, −7.85726112552936763605106326006, −6.33216172916735621184223258962, −5.79988087910084338697817371067, −5.220565337628569961126704852264, −4.15905206264333882937492846118, −3.29864281038136908596570178185, −2.339602154948460865072995653863, −0.92140223779599162401554456155, 0.55651189403082506702026674580, 1.66912816843950606443670644860, 2.42429375943152008331814990745, 3.84219858629885455060781558708, 4.77986623248907270641361375361, 5.364583607713960408260699600663, 6.637299879910382394366740204473, 7.16512609885646145273206683204, 7.67950257248032838201341913276, 8.835218532355195557146071629614, 9.77578293533899840884983324030, 10.73879369069274255304156881900, 11.28011822685152070837507440943, 12.05678075272067223219720592429, 12.91136245515333084848847596413, 13.62081385349492754982388354604, 14.15098003376749436408439958562, 15.25224365061331644076356154121, 16.14212188710015518065414825672, 16.90021440701761595732979984362, 17.53459729860189764119604624867, 18.07162610438401171303657912808, 19.006147256036671590869749788434, 19.56074410552114543875085776592, 20.53407674389527552288072426089