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Properties

Label 1-1021-1021.237-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.981 - 0.190i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1538624691 + 1.596369607i\)
\(L(\frac12)\) \(\approx\) \(-0.1538624691 + 1.596369607i\)
\(L(1)\) \(\approx\) \(1.101871964 + 0.9505559132i\)
\(L(1)\) \(\approx\) \(1.101871964 + 0.9505559132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.989 + 0.147i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (-0.0554 + 0.998i)T \)
7 \( 1 + (-0.763 - 0.645i)T \)
11 \( 1 + (-0.993 - 0.110i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
17 \( 1 + (-0.966 + 0.255i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (0.830 - 0.557i)T \)
29 \( 1 + (-0.763 + 0.645i)T \)
31 \( 1 + (-0.128 + 0.991i)T \)
37 \( 1 + (0.510 - 0.859i)T \)
41 \( 1 + (-0.982 - 0.183i)T \)
43 \( 1 + (0.237 + 0.971i)T \)
47 \( 1 + (-0.659 - 0.751i)T \)
53 \( 1 + (0.786 + 0.617i)T \)
59 \( 1 + (0.975 + 0.219i)T \)
61 \( 1 + (-0.659 - 0.751i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.903 + 0.429i)T \)
73 \( 1 + (-0.659 + 0.751i)T \)
79 \( 1 + (0.869 + 0.494i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (-0.602 + 0.798i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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