Properties

Label 1-199-199.25-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.998 - 0.0628i$
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  + (0.0475 − 0.998i)2-s + (−0.786 + 0.618i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.580 + 0.814i)6-s + (0.928 − 0.371i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.723 + 0.690i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (0.0475 − 0.998i)13-s + (−0.327 − 0.945i)14-s + (0.0475 − 0.998i)15-s + (0.981 + 0.189i)16-s + (−0.959 + 0.281i)17-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (−0.786 + 0.618i)3-s + (−0.995 − 0.0950i)4-s + (−0.654 + 0.755i)5-s + (0.580 + 0.814i)6-s + (0.928 − 0.371i)7-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.723 + 0.690i)10-s + (−0.959 + 0.281i)11-s + (0.841 − 0.540i)12-s + (0.0475 − 0.998i)13-s + (−0.327 − 0.945i)14-s + (0.0475 − 0.998i)15-s + (0.981 + 0.189i)16-s + (−0.959 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.998 - 0.0628i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.998 - 0.0628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007286800617 - 0.2316629505i\)
\(L(\frac12)\) \(\approx\) \(0.007286800617 - 0.2316629505i\)
\(L(1)\) \(\approx\) \(0.4988042730 - 0.1986897381i\)
\(L(1)\) \(\approx\) \(0.4988042730 - 0.1986897381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.0475 - 0.998i)T \)
3 \( 1 + (-0.786 + 0.618i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.928 - 0.371i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.0475 - 0.998i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (-0.888 - 0.458i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.235 - 0.971i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.928 - 0.371i)T \)
53 \( 1 + (0.0475 + 0.998i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (-0.786 - 0.618i)T \)
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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

−27.405119942805727687732610240, −26.48206190311320011701369848576, −25.14501459744982946780863021573, −24.18199522531876629234544037324, −23.95475304238659332856960673898, −23.09813091550928059879495525277, −21.92186946150575181241018408645, −20.97159073216779693484085030248, −19.41697989613918606844567682363, −18.374660185547397301417053555607, −17.825172835200787645959705319453, −16.525023500545488455735761234889, −16.16480701570845912974145185545, −14.91904277815374482390480830170, −13.72792085688357089456986711184, −12.75781004631641866595343836805, −11.87027054562117918156464602307, −10.77810932988968641086203944308, −8.99789353256024159704437011604, −8.09502656878146489162406741440, −7.33292195488052963805641920800, −5.99165799175117065992756419049, −5.06634745958928444934522891869, −4.226082338287357105026827037224, −1.71558647385874264955081079135, 0.19209851887449814057143802617, 2.270358218405334886932048520879, 3.723964988894071098190088070519, 4.584982160483613411423445252229, 5.66896059057706527543073060941, 7.390143173910375670316333888976, 8.551563887733658407402868060188, 10.11213536840288067346806476994, 10.79017549610174417825862645404, 11.32283246231390463922551664969, 12.417326242243872063607695710606, 13.58622202419621926907883543611, 15.00252912348132114655572279241, 15.4977490581616667643776792193, 17.23863115668233798219103556504, 17.89252745113905212861533028932, 18.67940314098275973765915663070, 20.1346619578460826875496426473, 20.643455200111442911830454891924, 21.91105671654846286209619342647, 22.368458209366640054280248086155, 23.53225382775561399159090277160, 23.891067733305025895903298996524, 26.161451482334642268377616111202, 26.65951438626405543495793117652

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