Properties

Degree 1
Conductor 1021
Sign $-0.981 - 0.190i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.989 + 0.147i)2-s + (0.0922 + 0.995i)3-s + (0.956 + 0.291i)4-s + (−0.0554 + 0.998i)5-s + (−0.0554 + 0.998i)6-s + (−0.763 − 0.645i)7-s + (0.903 + 0.429i)8-s + (−0.982 + 0.183i)9-s + (−0.201 + 0.979i)10-s + (−0.993 − 0.110i)11-s + (−0.201 + 0.979i)12-s + (0.445 + 0.895i)13-s + (−0.659 − 0.751i)14-s + (−0.999 + 0.0369i)15-s + (0.830 + 0.557i)16-s + (−0.966 + 0.255i)17-s + ⋯
L(s,χ)  = 1  + (0.989 + 0.147i)2-s + (0.0922 + 0.995i)3-s + (0.956 + 0.291i)4-s + (−0.0554 + 0.998i)5-s + (−0.0554 + 0.998i)6-s + (−0.763 − 0.645i)7-s + (0.903 + 0.429i)8-s + (−0.982 + 0.183i)9-s + (−0.201 + 0.979i)10-s + (−0.993 − 0.110i)11-s + (−0.201 + 0.979i)12-s + (0.445 + 0.895i)13-s + (−0.659 − 0.751i)14-s + (−0.999 + 0.0369i)15-s + (0.830 + 0.557i)16-s + (−0.966 + 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.981 - 0.190i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.981 - 0.190i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.981 - 0.190i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (237, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.981 - 0.190i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1538624691 + 1.596369607i$
$L(\frac12,\chi)$  $\approx$  $-0.1538624691 + 1.596369607i$
$L(\chi,1)$  $\approx$  1.101871964 + 0.9505559132i
$L(1,\chi)$  $\approx$  1.101871964 + 0.9505559132i

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

−20.952238993254402927226906845689, −20.58258240332717491858079015925, −19.69173191725292372010529862622, −19.0797470525506766456253264725, −18.22068657192213005445808707222, −17.17166822757741969322657795610, −16.38514933725547125847076246983, −15.34147472892495551454132234037, −15.11430967577222901855399968277, −13.4513761214013187651461153571, −13.18934153546490348092342888131, −12.80696437263268644320695948391, −11.89254958166673919031490319608, −11.16673266252313279903881269566, −10.01935431034308079107447659283, −8.876749725816725746442065149530, −8.08406925412981876219538068274, −7.19863400523644387429615165607, −6.11944377154400912034138137083, −5.63916285983530459499718026694, −4.73141874248795809764969308533, −3.485241398007491888755358682358, −2.566861961958624047530726193148, −1.812215235242294991181039787554, −0.416130257504343389395574797290, 2.18250395836409897397072781663, 2.98050396545626877448045596449, 3.76904600175976225288459491121, 4.42697288604329403042100256693, 5.46329727677989373644366903562, 6.52811730227556862244227744288, 6.930115051718203257558645963224, 8.162222430047877878781489357886, 9.218791228614427642489015495311, 10.48312496748021981239750773797, 10.73506538624236914469951691259, 11.40480537777130963766700575157, 12.75147519186197841606565345490, 13.45545061645063695899645713857, 14.19779574261415214749069679354, 14.98291603743849237302685595796, 15.54993545244451567510505395091, 16.34722025351586206284210279189, 16.880192901503828942806180280725, 18.070962787772951648284537723738, 19.22281892278290660232815987098, 19.81769720366697840699015486933, 20.7192426168809850252619141265, 21.50775989871343338054321261253, 21.88863020991927220326587100540

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