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Properties

Label 1-203-203.121-r0-0-0
Degree $1$
Conductor $203$
Sign $0.517 + 0.855i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (−0.955 + 0.294i)2-s + (−0.0747 + 0.997i)3-s + (0.826 − 0.563i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.623 + 0.781i)8-s + (−0.988 − 0.149i)9-s + (−0.826 + 0.563i)10-s + (0.988 − 0.149i)11-s + (0.5 + 0.866i)12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + (0.365 − 0.930i)16-s + (0.5 − 0.866i)17-s + (0.988 − 0.149i)18-s + (−0.0747 − 0.997i)19-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (−0.0747 + 0.997i)3-s + (0.826 − 0.563i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.623 + 0.781i)8-s + (−0.988 − 0.149i)9-s + (−0.826 + 0.563i)10-s + (0.988 − 0.149i)11-s + (0.5 + 0.866i)12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + (0.365 − 0.930i)16-s + (0.5 − 0.866i)17-s + (0.988 − 0.149i)18-s + (−0.0747 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.517 + 0.855i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.517 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8030292187 + 0.4529315615i\)
\(L(\frac12)\) \(\approx\) \(0.8030292187 + 0.4529315615i\)
\(L(1)\) \(\approx\) \(0.7966780568 + 0.2985817957i\)
\(L(1)\) \(\approx\) \(0.7966780568 + 0.2985817957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.955 + 0.294i)T \)
3 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (0.733 + 0.680i)T \)
37 \( 1 + (0.988 + 0.149i)T \)
41 \( 1 - T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.988 + 0.149i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.955 + 0.294i)T \)
97 \( 1 + (0.900 + 0.433i)T \)
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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

−26.58540959110312507770904153919, −25.47730504722340680547748646352, −25.21184611634871623427156347301, −24.23403525086290042563904111568, −22.85690318200900417243447776254, −21.8976088874283180370530611842, −20.70306039523350738471272606974, −19.89389462149200872184222886927, −18.767295799093923813793003351680, −18.240604837313503242522210450006, −17.231804126981964682038085557211, −16.732713888706114235376917729642, −15.02014018179884181216879232569, −13.94128168078890871542948405978, −12.78992251903809177672096470764, −11.98874204356156258125724515763, −10.77497403021834808090338186551, −9.880867450329815909011085235343, −8.641074312841973508893324712715, −7.78314077480097825037539161940, −6.442012464400153523360402379707, −5.95804017971793641147139135215, −3.47441739197132769755629184835, −2.12939996443175537815865927145, −1.227228420243079064770466804982, 1.35635035108282195773528249497, 2.91781975461740705003376889283, 4.625361438584492113455821633645, 5.83718623489813624510800230573, 6.6976969138166234350828978811, 8.40636719591693936759802840262, 9.35228658824925224575427528061, 9.7518685290700772190452555451, 11.043989770519824915703816253855, 11.830124683094263185132098971025, 13.78527375265262343082640979560, 14.51321459342236408001196872216, 15.7922061593137643882410019985, 16.5018536439373396756528225728, 17.29517143968338778116416541877, 18.11289433465075217801817747467, 19.44152550981257552453258989570, 20.35412590841648846649739535246, 21.22846668571998306589138446045, 21.98818157194744363490303765382, 23.32700086079063877567936522301, 24.49135837707254778198042612405, 25.45914777079693508602402296271, 26.01894770648308482061791073169, 27.02920152192819603117571114898

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