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Properties

Label 1-199-199.52-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: $\chi_{199} (52, \cdot )$
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

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

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