↑

Properties

Label 1-211-211.171-r0-0-0
Degree $1$
Conductor $211$
Sign $-0.0473 + 0.998i$
Analytic cond. $0.979879$
Root an. cond. $0.979879$
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  + (0.623 + 0.781i)2-s + (0.623 + 0.781i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.222 + 0.974i)9-s + 10-s + (0.623 + 0.781i)11-s + (−0.900 + 0.433i)12-s + (−0.900 − 0.433i)13-s + 14-s + 15-s + (−0.900 − 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.623 + 0.781i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.222 + 0.974i)9-s + 10-s + (0.623 + 0.781i)11-s + (−0.900 + 0.433i)12-s + (−0.900 − 0.433i)13-s + 14-s + 15-s + (−0.900 − 0.433i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0473 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0473 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $-0.0473 + 0.998i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ -0.0473 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.452080862 + 1.522564496i\)
\(L(\frac12)\) \(\approx\) \(1.452080862 + 1.522564496i\)
\(L(1)\) \(\approx\) \(1.489876700 + 0.9940351398i\)
\(L(1)\) \(\approx\) \(1.489876700 + 0.9940351398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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

−26.585157322017331635281303418692, −25.12854606436325414346687654276, −24.556395228292404265747808794619, −23.77669949006265350508464410217, −22.25622137187632695728388574855, −21.93492298188520605123178275116, −20.785923915146109188209499006765, −19.87242253350885641105657099178, −18.72849658624512085308741033688, −18.469250251244992125491537755626, −17.272365618324588456935348445959, −15.267314239524074028101545722191, −14.52831505949407115156310051504, −13.89305650667691509522257940885, −12.968976150748678312080126480786, −11.735603764961723626920417242189, −11.18366700795792984025227059095, −9.54375595323487539005877379586, −8.92519119362284065238364278368, −7.227388795943789446383607949236, −6.21726375699654967099093816960, −5.10273643212152316708981326054, −3.37747447299477807191069490867, −2.48043223696262457330198591080, −1.554719267486648377357578580425, 2.03858190980825388066928537363, 3.65616845934555715978949242809, 4.70689663544732598014437927512, 5.26118512866232607449973897194, 6.9762433227293937172888619873, 7.977558547444566182423183802699, 9.05762840668072309699498116182, 9.88983296482106970734089478793, 11.373622685196871972324124570485, 12.79781113327983078104110366050, 13.57190040245972184830311564359, 14.56667008993029993873940467529, 15.14704822194199558555037268267, 16.45761902162547536550281898146, 17.08505145517064703856672718904, 17.85938035249236767866807525491, 20.022431608146544320853719437423, 20.32836544908141323447785470341, 21.438266232526159572124315365335, 22.13041672568966090535037943723, 23.17007429390128152923340841921, 24.54734548961053204473498404232, 24.80221170509078753654642991779, 25.89374919081558984495799055661, 26.81012516445178119261098939293

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