Properties

Label 1-1480-1480.779-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: $\chi_{1480} (779, \cdot )$
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

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

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