↑

Properties

Label 1-403-403.295-r0-0-0
Degree $1$
Conductor $403$
Sign $0.782 - 0.622i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.978 + 0.207i)2-s + (0.669 − 0.743i)3-s + (0.913 − 0.406i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.669 − 0.743i)15-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.669 − 0.743i)3-s + (0.913 − 0.406i)4-s + 5-s + (−0.5 + 0.866i)6-s + (0.913 − 0.406i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.669 − 0.743i)15-s + (0.669 − 0.743i)16-s + (−0.104 − 0.994i)17-s + (0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ 0.782 - 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346777452 - 0.4705572128i\)
\(L(\frac12)\) \(\approx\) \(1.346777452 - 0.4705572128i\)
\(L(1)\) \(\approx\) \(1.104174704 - 0.2226942666i\)
\(L(1)\) \(\approx\) \(1.104174704 - 0.2226942666i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.978 + 0.207i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + T \)
7 \( 1 + (0.913 - 0.406i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.669 + 0.743i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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

−24.65318846171466661594936225439, −24.13508064292299566426155242358, −22.03847680153034910908802989027, −21.69227615292862252250788303004, −20.84238074943925478360820113771, −20.31800112794935275190814035942, −19.07034078279703086687856173768, −18.497760277224719498983447211043, −17.3080411997957945411007339312, −16.84962707800454311789750694223, −15.65783778948553754926398944394, −14.92270640996589352012860719656, −13.9362111402328476434393244222, −12.969268587063507611217405342363, −11.46741036000341283467947991805, −10.76630677778487030603417441301, −9.97145755394439818124550580007, −8.80787278890810534722273112484, −8.64197679054407640920054348682, −7.373834036229742161578378090234, −5.97972117816754721790240425329, −5.006769271035291368840343050211, −3.41709084350235271749752918312, −2.44627311736666357363420386203, −1.48325521918791488187431420854, 1.28875854780046709417573840146, 1.906045038649552991362882249355, 3.024280023657125008840987493347, 4.95380251884584061816579223030, 6.08938707468062641654168966611, 7.3262251850409292651325438021, 7.58601561015816373760167722076, 8.977121213369225718737571514812, 9.502599173653489757184541449532, 10.55635679755840763464819373003, 11.647388202152021692215502829779, 12.69371048493647841046373936257, 13.8656207921500339639367550775, 14.49128645243014603046556701960, 15.35221132763631277601693071103, 16.724170375096791461660070938129, 17.57686661848598542285916710709, 18.07682078985938323898886415187, 18.75677539548502600565306184142, 19.97392846546012143362359566495, 20.66775625517812142020332235639, 21.05275836101786719147399366027, 22.73816205735109271902712281734, 23.77165985237780911770770547267, 24.68928826145053660251606026398

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