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Properties

Label 1-1480-1480.19-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.934 - 0.355i$
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  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)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 + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)27-s + (0.866 − 0.5i)29-s − i·31-s + (0.939 − 0.342i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)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 + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)27-s + (0.866 − 0.5i)29-s − i·31-s + (0.939 − 0.342i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.772602808 - 0.3257287021i\)
\(L(\frac12)\) \(\approx\) \(1.772602808 - 0.3257287021i\)
\(L(1)\) \(\approx\) \(1.213452590 - 0.2284286046i\)
\(L(1)\) \(\approx\) \(1.213452590 - 0.2284286046i\)

Euler product

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

−20.82143425296172585617379922670, −19.90223223715003609988112378463, −19.58794769841687520679602853243, −18.3200785407056812253227339610, −17.66620698582533929890070099741, −16.62171630776670475215140751071, −16.39606944080163279746373189376, −15.47836607383342775826006277075, −14.52621945776628863157108516291, −13.966094845183343088058970523021, −13.56463381642288208167611575092, −11.91013521249604638829890166051, −11.45888917017252486008339625336, −10.83470113150898080792258941515, −9.764060869488240423421527485476, −9.29774774187506432757816237582, −8.35981404921940598026542818119, −7.59344636841569222990575236521, −6.58691803496052642798757266855, −5.52894977796752828029514594158, −4.71503281578079611357840572801, −4.06303092952383258312721975653, −3.19819137845597173992401797374, −2.13806612235650644181068060927, −0.8316114286752959325443931494, 0.990425585150812127751974947410, 1.944635548598233673768905610678, 2.601826282983984393453915106982, 3.77329145837757663344095559484, 4.91907252055198475899915528181, 5.725298815313644788726972795606, 6.54583646178410615664438084669, 7.46591389169201511320590088988, 8.13414682257569024190232214457, 8.761904320242777684065709226688, 9.79135756198914225580366951411, 10.7115401251435073181862489207, 11.749160222218725220319066624015, 12.26946932577196429951921374792, 12.8075493715667678136749728781, 13.90586987168074731691257863238, 14.50337416477508235584021665172, 15.14944617114129393970352565925, 16.00676865442369431571116993320, 17.32474615106391276615196388464, 17.6817449531668062135602205542, 18.1972421199226593356484756463, 19.14514088443464770471024712229, 19.887911548076644826450934400079, 20.3747728626501434449377750725

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