↑

Properties

Label 1-4760-4760.157-r0-0-0
Degree $1$
Conductor $4760$
Sign $0.730 - 0.682i$
Analytic cond. $22.1053$
Root an. cond. $22.1053$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s − i·13-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s − i·29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.5 − 0.866i)37-s + (0.866 − 0.5i)39-s − i·41-s − i·43-s + (0.866 + 0.5i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s − i·13-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s − i·29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.5 − 0.866i)37-s + (0.866 − 0.5i)39-s − i·41-s − i·43-s + (0.866 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ 0.730 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017563670 - 0.4012227782i\)
\(L(\frac12)\) \(\approx\) \(1.017563670 - 0.4012227782i\)
\(L(1)\) \(\approx\) \(0.9978091238 + 0.2202852217i\)
\(L(1)\) \(\approx\) \(0.9978091238 + 0.2202852217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.323238543172882828205252756747, −17.77424261610938634352356725578, −16.8826681128260779153984798942, −16.28329598428691899145742351045, −15.49307363012465573652607063152, −14.73119161827864507114103377984, −14.147464601618927355223178135088, −13.46212237542771343246996140492, −13.00314229983702302164048184485, −12.22662099386264584822712008307, −11.58341857208666827724709520329, −10.86224156547751801566062160839, −10.05988120567827263554922187829, −9.13932414816386483658605353728, −8.59298486359762219900670586987, −8.00108802659505417780693675195, −7.21184536139877956040981963846, −6.50998507051462008051761242894, −6.0264658912360533832161516387, −4.891299877079687495902359911749, −4.267972648039002307931641193229, −3.07018052936091942420839751518, −2.69349530688188231342935078537, −1.77764931164113482666921668458, −0.93582886110065931128936249222, 0.28971412042470924730392673447, 1.81018220799235181034367127036, 2.49914044511959995872115304944, 3.25806721930879200123131595330, 4.05011860016064951764837011724, 4.66672969727785054568683399472, 5.612898520612189531354441823215, 5.951948613386787380809916876205, 7.34688722191104383524724221057, 7.91124469628828938632752853703, 8.35868230190129458152535706834, 9.42563599100474753267033046188, 9.83970463922365649317026811259, 10.60787681800673349300706382965, 10.966443787534316068205437055410, 12.127024798042440160876802779214, 12.6344774611552593580334872308, 13.65274573705657983920901850160, 13.90676734402367139105883506395, 15.05866623625553787364861267191, 15.300327644156581921442457589546, 15.85856088074047606563466630654, 16.674717433729269549609214132009, 17.38259960769578971045046077423, 17.940617357261608387141327772201

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