Properties

Degree 1
Conductor 1021
Sign $-0.999 + 0.0166i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.886 − 0.462i)2-s + (0.0922 − 0.995i)3-s + (0.572 + 0.819i)4-s + (−0.542 + 0.840i)5-s + (−0.542 + 0.840i)6-s + (0.997 − 0.0738i)7-s + (−0.128 − 0.991i)8-s + (−0.982 − 0.183i)9-s + (0.869 − 0.494i)10-s + (−0.412 − 0.911i)11-s + (0.869 − 0.494i)12-s + (0.445 − 0.895i)13-s + (−0.918 − 0.395i)14-s + (0.786 + 0.617i)15-s + (−0.343 + 0.938i)16-s + (−0.0554 − 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.886 − 0.462i)2-s + (0.0922 − 0.995i)3-s + (0.572 + 0.819i)4-s + (−0.542 + 0.840i)5-s + (−0.542 + 0.840i)6-s + (0.997 − 0.0738i)7-s + (−0.128 − 0.991i)8-s + (−0.982 − 0.183i)9-s + (0.869 − 0.494i)10-s + (−0.412 − 0.911i)11-s + (0.869 − 0.494i)12-s + (0.445 − 0.895i)13-s + (−0.918 − 0.395i)14-s + (0.786 + 0.617i)15-s + (−0.343 + 0.938i)16-s + (−0.0554 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0166i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0166i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.999 + 0.0166i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (12, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1021,\ (0:\ ),\ -0.999 + 0.0166i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.005038020521 - 0.6043648133i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.005038020521 - 0.6043648133i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5337618851 - 0.3640108928i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5337618851 - 0.3640108928i\)

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

−21.58295440675650907131334078428, −21.09203339087600794239010260199, −20.380725892811922248147490816879, −19.72231404217938454345633345050, −18.94793996812749640873660846715, −17.78444708993464398982310810345, −17.17692952724176859076829498026, −16.62434387704525476421180245196, −15.583490423283085976943045499949, −15.29846203091736237498737008631, −14.51873901727221406161323811752, −13.485938692364708788518321769692, −12.078047711362255394297842051170, −11.43036235235824020349428335483, −10.64758366885030279684809591194, −9.813426980354314276314958454526, −8.96765484747949852055396871658, −8.30072164107563709751821468375, −7.80415853189504442114445006441, −6.485201773551090937511902260, −5.43144111368212336877606404333, −4.62647715855825308663413430908, −4.03265457245132318264971095700, −2.306880192632144184288940078855, −1.41312937898299695859143051498, 0.35587364314230254782583443322, 1.398128925021239316448745424149, 2.688964558500757133796103362099, 2.99799382718305794231844813685, 4.40711148659089679613772839549, 5.93727340707463158958729516154, 6.73481841951147184064542048682, 7.623793556172104447678489774346, 8.26891862844195727862679014149, 8.63256374593006167404698596580, 10.272351555088852267475908952960, 10.852516927027303343491276916363, 11.5308984153293867028739441769, 12.17373819114137433795984883557, 13.18245039211726193288531401375, 14.03103767546422912034411015045, 14.8827313207001392493045819727, 15.83612819055858626511228494915, 16.73637173369772460783327308248, 17.77434950426493500007471917914, 18.23080527461149679221974026146, 18.66766336294841762392554076777, 19.56108050300249889640114226226, 20.16180001358350005257492209649, 21.02418305755211226987175633488

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