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Properties

Label 1-1480-1480.1403-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.375 + 0.926i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.375 + 0.926i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.375 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9062900924 + 1.345290517i\)
\(L(\frac12)\) \(\approx\) \(0.9062900924 + 1.345290517i\)
\(L(1)\) \(\approx\) \(1.198365817 + 0.4031406058i\)
\(L(1)\) \(\approx\) \(1.198365817 + 0.4031406058i\)

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

−20.40967475638340890334869813243, −19.88730675835828948097526216065, −19.00422653583765200698563921553, −18.23956189410624880681400237223, −17.598354885574055672275485039900, −16.574041515670039256051701644525, −15.69756076233630027839414962632, −15.20880285369539429612695654398, −14.34854621661195624211311619449, −13.43909054237220359178692138614, −13.02087833251307010993219198085, −12.39142900321528116888487175578, −10.98630912694955679884617360259, −10.39060542942777215462748088671, −9.48703821097254992345806204085, −8.92896449432190636924476139774, −7.878984347754012152457424930, −7.14210642217125341425882684629, −6.71393478136221375023892587631, −5.23703177687743514504792412606, −4.45169723867767325345125645078, −3.37730316555615958168053418132, −2.831777974113658397949415543365, −1.76670686950322226994281659780, −0.501454232581264215816974247150, 1.543364902249438142488393585905, 2.326406858945280284358238786550, 3.37001832768475625744435028416, 3.838553331358040788738658861096, 5.074204472345536618784582008581, 6.04170426346343942651892665912, 6.78147754916050379575532956176, 7.95811145991751082237274755682, 8.51218426222985469579574425678, 9.25715863653935900846409929794, 9.91065889969919956927677667425, 10.89275833456252066536150494666, 11.70313525188222974786200906312, 12.94131218880388037928807420492, 13.12563574698103108079200477223, 14.157351794427264538565346314415, 14.86984815918457644856549713022, 15.5581157997085614072927930178, 16.27306337248407891659268312254, 16.89604521067190383590970504996, 18.26316972740435538862326025743, 18.865168765270175791646192899023, 19.238266395366341155329572777888, 20.06558624710078437545035723333, 21.15254004343641866258702162474

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