Properties

Degree 1
Conductor 139
Sign $-0.632 + 0.774i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.377 + 0.926i)2-s + (−0.158 − 0.987i)3-s + (−0.715 + 0.699i)4-s + (0.291 + 0.956i)5-s + (0.854 − 0.519i)6-s + (−0.877 + 0.480i)7-s + (−0.917 − 0.398i)8-s + (−0.949 + 0.313i)9-s + (−0.775 + 0.631i)10-s + (−0.419 + 0.907i)11-s + (0.803 + 0.595i)12-s + (0.934 + 0.356i)13-s + (−0.775 − 0.631i)14-s + (0.898 − 0.439i)15-s + (0.0227 − 0.999i)16-s + (0.538 + 0.842i)17-s + ⋯
L(s,χ)  = 1  + (0.377 + 0.926i)2-s + (−0.158 − 0.987i)3-s + (−0.715 + 0.699i)4-s + (0.291 + 0.956i)5-s + (0.854 − 0.519i)6-s + (−0.877 + 0.480i)7-s + (−0.917 − 0.398i)8-s + (−0.949 + 0.313i)9-s + (−0.775 + 0.631i)10-s + (−0.419 + 0.907i)11-s + (0.803 + 0.595i)12-s + (0.934 + 0.356i)13-s + (−0.775 − 0.631i)14-s + (0.898 − 0.439i)15-s + (0.0227 − 0.999i)16-s + (0.538 + 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.632 + 0.774i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.632 + 0.774i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $-0.632 + 0.774i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (81, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ -0.632 + 0.774i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3912070154 + 0.8247909310i$
$L(\frac12,\chi)$  $\approx$  $0.3912070154 + 0.8247909310i$
$L(\chi,1)$  $\approx$  0.7951803554 + 0.5532898590i
$L(1,\chi)$  $\approx$  0.7951803554 + 0.5532898590i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.1302697265018388550916745346, −27.42634446552828448243578250913, −26.35469845806549446733693636866, −25.19103609267531715556336322575, −23.59995702911147200069819413714, −23.01845881807751488826704757208, −21.83522312035781791711143947529, −21.02890030640426425368360496064, −20.37672108675055173639683694640, −19.39546662226832320840482134622, −18.07459046974979755411628977415, −16.6236917514389198221808106194, −16.07471120381395883742664452449, −14.59478825271289392487402838979, −13.392705050753362834150021671738, −12.70182388984854973703059684525, −11.22241696886359231554330609421, −10.43468886768791420634805697943, −9.35034773101149942050625930162, −8.5839655645758219939173383158, −6.06301795602881904337491914769, −5.15328564161231030786773435974, −3.949803853308057942425061360527, −2.95095680114805054956555999973, −0.740241002521091117761921140, 2.2967904804468263246928200569, 3.653782195558928716385389987850, 5.64299805143313537520931306471, 6.40055333222331634701385696587, 7.21290900609787961527351824687, 8.4034175569457740094610024278, 9.782256435819880850773151523093, 11.37732999189266124387913867983, 12.79743753143060912593835040021, 13.228887324766669918218732320167, 14.56172465140246222421836321249, 15.32625254808476001495168993927, 16.69491767813355792825147046828, 17.69123312355918901097029899044, 18.58196915710042786637366526997, 19.23582549037908424319581552157, 21.12157388161914300639324781454, 22.22063933487145146993071429849, 23.1508740126251144546546283466, 23.54340692211177006845228999053, 25.07681398132079302477122135875, 25.69879685535522084595759419921, 26.12788542994830219214503157552, 27.82108430917335067983077667590, 28.885938868762458499018944154831

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