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Properties

Label 1-1480-1480.83-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.495 + 0.868i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.495 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6159335717 + 0.3579684895i\)
\(L(\frac12)\) \(\approx\) \(0.6159335717 + 0.3579684895i\)
\(L(1)\) \(\approx\) \(0.8427613172 - 0.1541694611i\)
\(L(1)\) \(\approx\) \(0.8427613172 - 0.1541694611i\)

Euler product

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

−20.49696102897369278691284084755, −19.860353580983360528495571334668, −19.10928521233931551896904213332, −18.55550390121327215226089215897, −17.2018885623587223747355957670, −16.6471881884016428752224068102, −16.00744961546000825521595453602, −15.45182097132117385356756242051, −14.36347967629797035400222692821, −13.934244340491106908856191082947, −13.01590222209872800647452529664, −12.128520123522566902802191580487, −11.16476992223818300116675744505, −10.453486050361681336553138166040, −9.6398507102812916617089113989, −9.2085500070352068603877274150, −8.14919239691615879398092346006, −7.42196133937980928584626181235, −6.187501876339401960356164098421, −5.58221272524564562143929332672, −4.523029234812560228205749978436, −3.66184219084908118105397046284, −3.038285925125493855977124903766, −2.043710782893452011618456538707, −0.27163384985469823596560218118, 1.0742886463794833383867702841, 2.26520659264773977942193360001, 2.95605336437897718346730092952, 3.77317535351268289286620494813, 5.3071255211582273455987996260, 5.75941451320407959919660003756, 7.00978241363732358099838807514, 7.40996284403393871530049555768, 8.14862369547385369288068325333, 9.473093518079088641001049751423, 9.65393531004841043198336822556, 10.8198463705875283023789359459, 12.034801247740305890875435446061, 12.43177034113880911824182082468, 13.058793096049806022073530336025, 13.91994773496835332030582963865, 14.62465455572008577793808654637, 15.502293216860341359639780521593, 16.23597066091161101056420743009, 17.14369680604329073107667832658, 18.14183907696456433331408296584, 18.34370439744249477267767487445, 19.3245731864464337088444906053, 20.17666790996186023009075019297, 20.28728533803584287040930029071

Graph of the $Z$-function along the critical line