Properties

Label 1-209-209.71-r1-0-0
Degree $1$
Conductor $209$
Sign $0.00483 + 0.999i$
Analytic cond. $22.4601$
Root an. cond. $22.4601$
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  + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.438 − 0.898i)5-s + (0.961 + 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.882 + 0.469i)15-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.438 − 0.898i)5-s + (0.961 + 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.882 + 0.469i)15-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00483 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00483 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.00483 + 0.999i$
Analytic conductor: \(22.4601\)
Root analytic conductor: \(22.4601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 209,\ (1:\ ),\ 0.00483 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6617799918 + 0.6649869282i\)
\(L(\frac12)\) \(\approx\) \(0.6617799918 + 0.6649869282i\)
\(L(1)\) \(\approx\) \(0.9559589656 - 0.08658261782i\)
\(L(1)\) \(\approx\) \(0.9559589656 - 0.08658261782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
3 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (0.438 - 0.898i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (-0.997 + 0.0697i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.374 + 0.927i)T \)
53 \( 1 + (-0.438 - 0.898i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.438 + 0.898i)T \)
73 \( 1 + (-0.615 + 0.788i)T \)
79 \( 1 + (-0.961 + 0.275i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.241 - 0.970i)T \)
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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

−26.140194482065902247733807046042, −25.133425532915733097461941689440, −24.44633341022816565738715187188, −23.57105317787541646194635298086, −22.71638086747131280407189420329, −22.00204575016428392608219912284, −20.60148662609698691858977554239, −19.33407916979542935360989276362, −18.308940852778249296460602353452, −17.49080619981214209564273232148, −17.14267117588276447209284520064, −15.4034350042986880716097193417, −14.5823108794614714355694741744, −13.686300647184193693828654033136, −13.11647544010475147556254779022, −11.70189901546911097776498084922, −10.55374682325847952399584627720, −9.11724631877101722095343684851, −7.684314943795078460898682295, −7.320973533134985118445785899664, −6.199647210981747115775119557165, −5.2171534613247796274992581419, −3.61385725118613346575332699610, −2.131016158017934588993550101261, −0.281327478828413856308576440981, 1.633133429957148294317810813896, 2.79960966057081605471824607434, 4.507385635708892942437026896713, 4.83023444948920105448658936014, 6.03780825319588834125658026628, 8.514631112591705042080141394530, 9.0342132847957648975133667162, 9.96501931790190420633195985973, 11.13310347508765026763355515978, 11.92389194616997579413924443469, 12.9627244946296029673862201159, 14.203903297998023526452859591580, 14.930738307068357945364845127850, 16.21223761995563842187589679676, 17.2698392594519395626918588000, 18.14131926155761401403970606029, 19.48950553615599906753801937815, 20.400094698974209764668579364318, 21.13623070543060076295189576991, 21.749571845791753070997578520, 22.511012662640347851145176228601, 23.89801001155936120908253817718, 24.646402645462418535037702516675, 26.06068368468386312653386268301, 27.08361591004376381043358619900

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