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Properties

Label 1-2280-2280.1493-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.942 - 0.334i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  − i·7-s + 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s − i·37-s + (0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s − 49-s + (−0.866 − 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  − i·7-s + 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + 31-s − i·37-s + (0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 − 0.5i)47-s − 49-s + (−0.866 − 0.5i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.023278604 - 0.3480606384i\)
\(L(\frac12)\) \(\approx\) \(2.023278604 - 0.3480606384i\)
\(L(1)\) \(\approx\) \(1.274282416 - 0.1197389484i\)
\(L(1)\) \(\approx\) \(1.274282416 - 0.1197389484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
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 \( 1 + (0.5 + 0.866i)T \)
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 \( 1 + (-0.866 + 0.5i)T \)
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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

−19.44769601974227374648343719166, −19.10851026805639906193833683906, −18.24704330037700970282086120422, −17.627033244832312750814437979025, −16.81707782932921863764304126667, −16.06285995227827067666092824283, −15.381059346583674042164861562219, −14.62796822888309410711853455564, −14.097734451970080748271566152121, −13.00384512051984317678574107393, −12.43612188994386232254612486806, −11.75842742747890510004955670331, −10.96484794476337779528227545442, −10.19178728623292734858301208716, −9.23627195527740290487612533272, −8.676275351892631675002830510294, −8.02053397038050076966386378480, −6.9011364548091033154528658023, −6.17838012814732005503292984407, −5.52684062668370561491177724966, −4.627226426263462733774739260409, −3.57148990000202396370919795682, −2.95284812495120477163206381386, −1.81516292746919700295982257999, −0.96260881529489141733978770184, 0.96047272872888181020277709147, 1.466269285848937289212794857724, 2.910792324482121743208519648204, 3.668980131792993776367320968895, 4.3936510569026170337827033586, 5.22717746682818390398054984243, 6.52793441511209441575603047346, 6.658615581783006139529778812424, 7.83776314120703825254918657515, 8.40777392830163409195287688701, 9.57472008200388407063623357907, 9.859466975086483423691288272760, 11.020994614560068549396378341067, 11.49210137826782272157276883444, 12.25318040378537374756040026253, 13.39117196735781567751256219675, 13.70904855020706202453007624946, 14.49887963489105229970823524367, 15.2411657331879353380763246858, 16.2489105250960902917810207384, 16.71489887274420974987251797823, 17.37386342965728606539569433682, 18.13918691868133714492107710863, 19.08927816405413413053482629595, 19.505505258349734125130872662625

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