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Properties

Label 1-199-199.111-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.956 + 0.290i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (−0.888 + 0.458i)2-s + (0.928 + 0.371i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (−0.995 + 0.0950i)6-s + (−0.786 − 0.618i)7-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (0.235 − 0.971i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (−0.888 + 0.458i)13-s + (0.981 + 0.189i)14-s + (−0.888 + 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.959 + 0.281i)17-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (0.928 + 0.371i)3-s + (0.580 − 0.814i)4-s + (−0.654 + 0.755i)5-s + (−0.995 + 0.0950i)6-s + (−0.786 − 0.618i)7-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (0.235 − 0.971i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (−0.888 + 0.458i)13-s + (0.981 + 0.189i)14-s + (−0.888 + 0.458i)15-s + (−0.327 − 0.945i)16-s + (−0.959 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.956 + 0.290i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07395832091 + 0.4985455267i\)
\(L(\frac12)\) \(\approx\) \(0.07395832091 + 0.4985455267i\)
\(L(1)\) \(\approx\) \(0.5374364608 + 0.3477847678i\)
\(L(1)\) \(\approx\) \(0.5374364608 + 0.3477847678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (0.928 + 0.371i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.786 - 0.618i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.888 + 0.458i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.0475 + 0.998i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.723 + 0.690i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (-0.888 - 0.458i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.981 - 0.189i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (0.928 - 0.371i)T \)
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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

−26.56088883029112722668701213878, −25.70434926167914912262871736839, −24.704517689343894970112901781830, −24.17246656858628208968014127791, −22.57310477795019062841570594023, −21.30182258158380676308385073440, −20.53450846123177596989885057740, −19.547278603174101344100936909735, −19.19023678222424354588976146917, −18.12795056118458167380796764411, −16.97725098924652820857933254454, −15.588745738472060927434498147359, −15.4408650248745237263004947219, −13.32953316152068663022709227463, −12.72747720310390645357331012822, −11.86691336315879916150111414179, −10.40695024177828838945921551634, −9.251919652907952142079467100357, −8.59563245833117687273122730351, −7.711690036651921523782342143152, −6.621814654923777861796016159387, −4.613345522372971868498878535749, −3.06845955298918067602128131314, −2.35192438301119321109385016955, −0.42607029140767744043969273975, 2.1544178149313276585213139164, 3.291101135179019279749442621308, 4.69919106350110626824200557083, 6.586127804085444466336228487384, 7.40193549917604587154801174332, 8.19936812085761009486077129131, 9.511171914517091356796501757202, 10.248013139602308498389684237206, 11.09554568987378805455477469625, 12.81145969264765498598743602315, 14.11618481454016784126409405174, 14.98587928762885481596284028357, 15.731362493130735820457607698514, 16.547558248906574823933881344016, 17.86340554659195204502478654535, 19.04529185067695038708542281878, 19.468228035286703265481814504325, 20.292088236180133645886777346473, 21.493011774446680350899620467869, 22.84703978179097089043512567157, 23.69145539465345700541268549765, 24.81031915718091664492379900463, 25.86563848215066587989894047319, 26.44090667373867546350855667328, 26.89417224118543473536688934990

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