↑

Properties

Label 1-767-767.42-r1-0-0
Degree $1$
Conductor $767$
Sign $0.923 + 0.383i$
Analytic cond. $82.4255$
Root an. cond. $82.4255$
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  + (−0.989 − 0.143i)2-s + (0.403 − 0.915i)3-s + (0.958 + 0.284i)4-s + (0.647 + 0.762i)5-s + (−0.530 + 0.847i)6-s + (0.874 + 0.484i)7-s + (−0.907 − 0.419i)8-s + (−0.674 − 0.738i)9-s + (−0.530 − 0.847i)10-s + (0.891 + 0.452i)11-s + (0.647 − 0.762i)12-s + (−0.796 − 0.605i)14-s + (0.958 − 0.284i)15-s + (0.837 + 0.546i)16-s + (−0.0180 + 0.999i)17-s + (0.561 + 0.827i)18-s + ⋯
L(s)  = 1  + (−0.989 − 0.143i)2-s + (0.403 − 0.915i)3-s + (0.958 + 0.284i)4-s + (0.647 + 0.762i)5-s + (−0.530 + 0.847i)6-s + (0.874 + 0.484i)7-s + (−0.907 − 0.419i)8-s + (−0.674 − 0.738i)9-s + (−0.530 − 0.847i)10-s + (0.891 + 0.452i)11-s + (0.647 − 0.762i)12-s + (−0.796 − 0.605i)14-s + (0.958 − 0.284i)15-s + (0.837 + 0.546i)16-s + (−0.0180 + 0.999i)17-s + (0.561 + 0.827i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 767 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 767 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(767\)    =    \(13 \cdot 59\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(82.4255\)
Root analytic conductor: \(82.4255\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{767} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 767,\ (1:\ ),\ 0.923 + 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.114357251 + 0.4210991308i\)
\(L(\frac12)\) \(\approx\) \(2.114357251 + 0.4210991308i\)
\(L(1)\) \(\approx\) \(1.093153951 - 0.05213695573i\)
\(L(1)\) \(\approx\) \(1.093153951 - 0.05213695573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.989 - 0.143i)T \)
3 \( 1 + (0.403 - 0.915i)T \)
5 \( 1 + (0.647 + 0.762i)T \)
7 \( 1 + (0.874 + 0.484i)T \)
11 \( 1 + (0.891 + 0.452i)T \)
17 \( 1 + (-0.0180 + 0.999i)T \)
19 \( 1 + (0.700 - 0.713i)T \)
23 \( 1 + (0.436 + 0.899i)T \)
29 \( 1 + (-0.619 - 0.785i)T \)
31 \( 1 + (-0.267 + 0.963i)T \)
37 \( 1 + (0.0901 - 0.995i)T \)
41 \( 1 + (0.997 - 0.0721i)T \)
43 \( 1 + (0.891 - 0.452i)T \)
47 \( 1 + (-0.647 + 0.762i)T \)
53 \( 1 + (0.468 + 0.883i)T \)
61 \( 1 + (0.619 - 0.785i)T \)
67 \( 1 + (0.817 - 0.576i)T \)
71 \( 1 + (-0.983 - 0.179i)T \)
73 \( 1 + (-0.796 - 0.605i)T \)
79 \( 1 + (-0.994 - 0.108i)T \)
83 \( 1 + (0.947 + 0.319i)T \)
89 \( 1 + (0.619 + 0.785i)T \)
97 \( 1 + (0.922 + 0.386i)T \)
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

−21.85761894047557890381349038002, −20.93253745203668091013053945009, −20.50375709234003189209812322613, −20.00971960745609288230333445259, −18.884910724553676413575750183628, −17.947053381686206445888978974464, −17.11026889076362626653214969808, −16.51356660033591559336033168164, −16.05774719286607073727144953173, −14.62226060347809732027543267063, −14.411524635868805211832323345113, −13.28921875770596870085257987715, −11.81264021218401715378711854602, −11.21864406379583806707710277557, −10.22945086740394111494749716320, −9.53319983146621984481194220629, −8.83740481295253194446758134113, −8.17943222335433824922607479306, −7.17348872647270813276935551689, −5.87134679867487179619703316018, −5.08630157867285766211167115660, −4.05045678663125398143210262751, −2.76267907329792169622721271417, −1.62797973772449890855885952031, −0.69018377116750715365322520195, 1.13424001677632860906481294051, 1.84876950242687805501612164924, 2.59274317444559649449853617574, 3.69342832104860689779192341869, 5.584725867771660876056029233086, 6.36976299732642207693325858091, 7.27327042268112174064847710800, 7.82422625145087701451667270988, 9.068672796102584810316327817137, 9.34130728734686565800467185052, 10.71249224767265207794027708446, 11.4027438362061890616604186298, 12.16704156848654929283226864683, 13.12640033271296058115105427004, 14.28515964954279491543793017031, 14.78334721588147986561208834269, 15.635279867472855220981526103, 17.20596729648787161468539853729, 17.57944534257789443522303735539, 18.07205810157634830437464699885, 19.045674945293337612676428041784, 19.49883242014163086942393350380, 20.443864504237428198179629802397, 21.300923469245713097574363694946, 21.98295364575903642683829772660

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