Properties

 Label 1-280-280.187-r1-0-0 Degree $1$ Conductor $280$ Sign $-0.910 - 0.413i$ Analytic cond. $30.0901$ Root an. cond. $30.0901$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯
 L(s)  = 1 + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s − i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$280$$    =    $$2^{3} \cdot 5 \cdot 7$$ Sign: $-0.910 - 0.413i$ Analytic conductor: $$30.0901$$ Root analytic conductor: $$30.0901$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{280} (187, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 280,\ (1:\ ),\ -0.910 - 0.413i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1068181501 - 0.4929036306i$$ $$L(\frac12)$$ $$\approx$$ $$0.1068181501 - 0.4929036306i$$ $$L(1)$$ $$\approx$$ $$0.6889802340 - 0.1544277387i$$ $$L(1)$$ $$\approx$$ $$0.6889802340 - 0.1544277387i$$

Euler product

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

Imaginary part of the first few zeros on the critical line

−25.92673447921288610076438590307, −24.78851097235565389032480656443, −23.65901337433049901212115376986, −23.24091886378836211168504048658, −22.09128647859151496114972743554, −21.27464326202443012929432107890, −20.735245420603494226914256527934, −19.111802267500738684738498589587, −18.55842627233528209845673727892, −17.33760040622384528729245435303, −16.53648209279976973836056998845, −15.91235754018163342072143911835, −14.74879435475674686246107132918, −13.728334901940497464928112315124, −12.496314376324470876069382302825, −11.61236789118757472385037007891, −10.77545002329182342950367723893, −9.8178988609538678155497607022, −8.78884588164817186881605422227, −7.43402913082105408129150908659, −6.25003374250426426870863217246, −5.395541311452557840241878863720, −4.29298816798447801553070571009, −3.12251848664993946698000014927, −1.28732340565882654189743057539, 0.18787731942642221200142912318, 1.55596407272335668305734840413, 2.96445274674087308301386672460, 4.68230888320562073635912097428, 5.44547079047866766427347533083, 6.66612125936894954433205600596, 7.506757201022014323194243630755, 8.61664832417528990021509443504, 10.244107815097218567275203650309, 10.675106069613032795050140914592, 12.07755377529366530985956405274, 12.66124499867112159387512438220, 13.56802843319872056393774307950, 14.96664517279997765444199981378, 15.77642531159776357008610348211, 16.98996453238912716048103359320, 17.593178290994521442661262891222, 18.4737846503174806024846923588, 19.391402977692269044199189176465, 20.45948661170048461559568185666, 21.51851744568476551432055680786, 22.456751745991947795057077670076, 23.2945871945280924681716452581, 23.83476277222804275890421225028, 25.090405249392831509467126580626