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Properties

Label 1-2280-2280.1379-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.977 + 0.211i$
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  + 7-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)29-s − 31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + 49-s + (0.5 − 0.866i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + 7-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)29-s − 31-s + 37-s + (0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + 49-s + (0.5 − 0.866i)53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594232059 + 0.1703122302i\)
\(L(\frac12)\) \(\approx\) \(1.594232059 + 0.1703122302i\)
\(L(1)\) \(\approx\) \(1.104151979 + 0.03215687325i\)
\(L(1)\) \(\approx\) \(1.104151979 + 0.03215687325i\)

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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.61018600802983501809353058814, −18.91939179757249109629520459521, −18.050097704649636853132849666899, −17.53955901637725256850094640845, −16.97785777119143808725887723872, −15.88841697641110016565221740798, −15.21098570075768049077719054966, −14.78262045326495351959061486335, −13.81140434511726261781106127641, −13.06093456904238036135574839394, −12.49799544530342922537247674164, −11.47569209353262924251766310670, −10.85129868499059678879214663471, −10.278976707771617828694437688055, −9.28021259675601468461252669353, −8.4601537319722739445616123902, −7.667497076266777339778668177826, −7.31246786384044677851790041631, −5.86077009524881394587164695403, −5.446228591728108809121311205661, −4.550523245945663649358288603652, −3.71553600508616145407342013418, −2.5760943038421229578829331141, −1.94098486361197388397720460437, −0.71483234308975345525244823149, 0.80217228368252823303034537075, 2.06244657677867489885311824474, 2.55822308722544241380246479166, 3.78713911933019683742519991550, 4.86581098380312408586439027585, 5.03948993588241177775546049821, 6.24057370430427007203931322751, 7.20713719881911349998274095038, 7.70320834450385965859286466886, 8.655549177477489167794378150609, 9.28128853258165855595331313351, 10.21357095976991129906550371709, 11.11910477647582121466026578533, 11.42456424990403024545898602028, 12.484724904014122964570699463912, 13.126212297283134219630479251214, 14.01244715761450648356152634603, 14.64257233885554468318718221017, 15.21479249879346585465515973022, 16.3067426547772045169133164506, 16.62622428245837149716483271243, 17.71115064700213613200230222147, 18.27364325547558018439303579066, 18.724644862996268968405733348660, 19.83994948590225356820170902864

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