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Properties

Label 1-4800-4800.1013-r0-0-0
Degree $1$
Conductor $4800$
Sign $0.260 + 0.965i$
Analytic cond. $22.2911$
Root an. cond. $22.2911$
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  + (0.707 − 0.707i)7-s + (−0.0784 + 0.996i)11-s + (−0.0784 − 0.996i)13-s + (−0.809 − 0.587i)17-s + (−0.233 + 0.972i)19-s + (−0.453 + 0.891i)23-s + (0.522 − 0.852i)29-s + (0.809 + 0.587i)31-s + (−0.649 + 0.760i)37-s + (−0.453 − 0.891i)41-s + (−0.382 + 0.923i)43-s + (−0.809 + 0.587i)47-s − i·49-s + (−0.233 − 0.972i)53-s + (0.760 + 0.649i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (−0.0784 + 0.996i)11-s + (−0.0784 − 0.996i)13-s + (−0.809 − 0.587i)17-s + (−0.233 + 0.972i)19-s + (−0.453 + 0.891i)23-s + (0.522 − 0.852i)29-s + (0.809 + 0.587i)31-s + (−0.649 + 0.760i)37-s + (−0.453 − 0.891i)41-s + (−0.382 + 0.923i)43-s + (−0.809 + 0.587i)47-s − i·49-s + (−0.233 − 0.972i)53-s + (0.760 + 0.649i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(22.2911\)
Root analytic conductor: \(22.2911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4800,\ (0:\ ),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031170846 + 0.7901734835i\)
\(L(\frac12)\) \(\approx\) \(1.031170846 + 0.7901734835i\)
\(L(1)\) \(\approx\) \(1.025343754 + 0.05610946679i\)
\(L(1)\) \(\approx\) \(1.025343754 + 0.05610946679i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.0784 + 0.996i)T \)
13 \( 1 + (-0.0784 - 0.996i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.233 + 0.972i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (0.522 - 0.852i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.649 + 0.760i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.233 - 0.972i)T \)
59 \( 1 + (0.760 + 0.649i)T \)
61 \( 1 + (0.649 + 0.760i)T \)
67 \( 1 + (-0.233 + 0.972i)T \)
71 \( 1 + (0.987 + 0.156i)T \)
73 \( 1 + (-0.453 + 0.891i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (-0.972 - 0.233i)T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−18.00438251051426955865347144768, −17.38099324622215790986685297540, −16.63215363887444184513599528983, −15.99393638146532050172769342405, −15.29116789035834024802407963017, −14.68893918247412111523298641497, −13.92805006228002780053708610445, −13.457312656949859807656526498509, −12.49578446410344327902406394890, −11.91350597728101426959471179728, −11.13017079997816889633275635252, −10.82460441180661940467102489417, −9.77833821184514241488821964212, −8.84599789214484724836420114860, −8.602719314796092563325056257520, −7.89065619338199981673915319103, −6.68340212948027447783881495660, −6.43038474620115419713429578172, −5.404538674640298542310539782886, −4.74675571911715234276814066502, −4.07251697094080370843681738869, −3.06212898542217245863825141402, −2.24925344039457857958514347997, −1.63735067255475024737426810338, −0.36607206601732912330464285245, 0.99003684434052961540054029324, 1.7838552980500344457754626254, 2.6092346190558733104526496337, 3.58307664980066130279841442211, 4.35622454483627698896934226803, 4.95897398999061543630249627671, 5.68965536176622250417225774950, 6.726611681971109104936475134905, 7.24259096499693129255326457181, 8.127262073144702941963084004977, 8.39869401241037726917097310348, 9.75378571914198602349149034095, 10.02747184550809721444506714292, 10.75188277256754828878907265461, 11.624423258524407847995564442986, 12.0830472246944388849165214984, 13.0352372927262142411462544847, 13.52228019901283523248536599978, 14.31196175807724825780252737687, 14.89694012070757598680449141541, 15.617391668297287420627997263402, 16.1431815021311807411417254392, 17.27757535517470006596595868669, 17.52732882754716974817973882360, 18.02295060344448145089791305667

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