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Properties

Label 1-1480-1480.987-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.446 + 0.894i$
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  + (0.984 − 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.342 + 0.939i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.342 + 0.939i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (−0.642 − 0.766i)7-s + (0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.342 + 0.939i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (−0.342 + 0.939i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.446 + 0.894i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.446 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473873638 + 0.9112503747i\)
\(L(\frac12)\) \(\approx\) \(1.473873638 + 0.9112503747i\)
\(L(1)\) \(\approx\) \(1.284272893 + 0.1413202037i\)
\(L(1)\) \(\approx\) \(1.284272893 + 0.1413202037i\)

Euler product

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

−20.36201939231103554571600914327, −19.882036128414561842749246606101, −19.02294634234029062867554229183, −18.487567383497951736075563452237, −17.74456579477936379738083560644, −16.461948714316804605334880154510, −15.82737173308429564310581525829, −15.40135752600400938495862828727, −14.45166639784228991680820581719, −13.70336622598453112384586812520, −13.05188671422386951603053754804, −12.30182227941289529535201259903, −11.36565765246012679838538438578, −10.193990950751402743032260269980, −9.82261448784428496381270914942, −8.720292574533804535557853236378, −8.362809945445682932372804405220, −7.36515253379314257851968207038, −6.4888663526084860629080574621, −5.403903465750615753331289864271, −4.724409783219991556682385053241, −3.2551643773198044877492340301, −3.02446562087504295688537633312, −2.11321169159246141699155766861, −0.56044289263166018931853247674, 1.34005857387022714442895695773, 2.12739209491902374478424675675, 3.1704144381782528356147854834, 4.00746839540653100648331011631, 4.62113003906112543523157329219, 6.10782524301816383144622986062, 6.80327571248138503087565317791, 7.74277201401349434560033850027, 8.13144658996297841597610659167, 9.40583130772243026685797638388, 9.918788925524952128502211070636, 10.45982326641511687484025156828, 11.900728941941878434957320967266, 12.49879645582354391449730844627, 13.33763529169372227014017497611, 13.95344394302786352817998549257, 14.64324234642259775589769415281, 15.46674480857121079523370267837, 16.18376524425894811597913377982, 17.03098727576755976697202683015, 17.8469934217632614794400938051, 18.79147681557081999785513884323, 19.39175875434509395523683178458, 19.92084544661015217225775281865, 20.80498869401103621796675341447

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