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Properties

Label 1-211-211.125-r0-0-0
Degree $1$
Conductor $211$
Sign $0.765 - 0.643i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.753 − 0.657i)2-s + (0.858 + 0.512i)3-s + (0.134 − 0.990i)4-s + (0.858 − 0.512i)5-s + (0.983 − 0.178i)6-s + (0.753 + 0.657i)7-s + (−0.550 − 0.834i)8-s + (0.473 + 0.880i)9-s + (0.309 − 0.951i)10-s + (−0.995 + 0.0896i)11-s + (0.623 − 0.781i)12-s + (−0.963 − 0.266i)13-s + 14-s + 15-s + (−0.963 − 0.266i)16-s + (−0.0448 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.753 − 0.657i)2-s + (0.858 + 0.512i)3-s + (0.134 − 0.990i)4-s + (0.858 − 0.512i)5-s + (0.983 − 0.178i)6-s + (0.753 + 0.657i)7-s + (−0.550 − 0.834i)8-s + (0.473 + 0.880i)9-s + (0.309 − 0.951i)10-s + (−0.995 + 0.0896i)11-s + (0.623 − 0.781i)12-s + (−0.963 − 0.266i)13-s + 14-s + 15-s + (−0.963 − 0.266i)16-s + (−0.0448 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.319173146 - 0.8457277706i\)
\(L(\frac12)\) \(\approx\) \(2.319173146 - 0.8457277706i\)
\(L(1)\) \(\approx\) \(1.992666089 - 0.5489971382i\)
\(L(1)\) \(\approx\) \(1.992666089 - 0.5489971382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.753 - 0.657i)T \)
3 \( 1 + (0.858 + 0.512i)T \)
5 \( 1 + (0.858 - 0.512i)T \)
7 \( 1 + (0.753 + 0.657i)T \)
11 \( 1 + (-0.995 + 0.0896i)T \)
13 \( 1 + (-0.963 - 0.266i)T \)
17 \( 1 + (-0.0448 + 0.998i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.963 - 0.266i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.995 - 0.0896i)T \)
41 \( 1 + (0.936 + 0.351i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.473 - 0.880i)T \)
53 \( 1 + (0.134 + 0.990i)T \)
59 \( 1 + (0.936 + 0.351i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.550 + 0.834i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.393 - 0.919i)T \)
97 \( 1 + (-0.691 + 0.722i)T \)
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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

−26.251123597684909699393712735622, −25.93092666782360489833557908510, −24.75273708644838426259176113147, −24.157103807065288392749286476959, −23.30152187722024035238040152720, −22.07203813323772328570354527601, −21.097504807432507368954904842229, −20.58888134809703839024597295345, −19.21103263023293982210018379249, −17.858459290386072979665047189465, −17.525464069518680333649921841003, −16.03273526340038864770230577663, −14.8675742060900375686393485105, −14.193093119968561181404884877320, −13.55514874608150724125244098580, −12.68007968591281389547889765177, −11.31723384973223982224599330213, −9.94635828889414483809196541445, −8.6631534985905694351047757533, −7.46430697863025692705214086676, −7.00409448341313169752153078901, −5.56402868622648460958954511373, −4.40996102137546079099881318822, −2.923035270949995433005821273144, −2.07447363186640619086785734160, 2.0063505666501434061955245047, 2.410776013784547529304798015744, 4.09622518906946452371434702987, 5.058272529319418721624266583057, 5.89974262779202611165992914913, 7.89161155851916572568491623665, 8.93875672701973538050876637905, 10.046972466205655370163636660350, 10.65757435196811720478276665317, 12.289145790963788629578880796709, 12.99490820560066796491707925709, 14.04517668905984418174261787068, 14.84190360096753493780401630836, 15.56503486814216175059882163871, 17.0000093954026514366952723056, 18.31364393445803746925250369387, 19.2412993990980033201637331224, 20.398623638815307207504349216938, 20.99731285696852184326454192301, 21.595158388319540188632814361074, 22.41340364499781264554112383473, 24.04218013765462051264165412304, 24.53429941934924041070062314368, 25.496858155441308367484378942817, 26.56320929365528886633236046661

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