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Properties

Label 1-203-203.107-r0-0-0
Degree $1$
Conductor $203$
Sign $0.703 + 0.710i$
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  + (−0.988 − 0.149i)2-s + (−0.733 − 0.680i)3-s + (0.955 + 0.294i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (−0.900 − 0.433i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.5 − 0.866i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)15-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)17-s + (0.0747 − 0.997i)18-s + (−0.733 + 0.680i)19-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)2-s + (−0.733 − 0.680i)3-s + (0.955 + 0.294i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (−0.900 − 0.433i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.5 − 0.866i)12-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)15-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)17-s + (0.0747 − 0.997i)18-s + (−0.733 + 0.680i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2727748854 + 0.1138235499i\)
\(L(\frac12)\) \(\approx\) \(0.2727748854 + 0.1138235499i\)
\(L(1)\) \(\approx\) \(0.4108467876 - 0.05270937326i\)
\(L(1)\) \(\approx\) \(0.4108467876 - 0.05270937326i\)

Euler product

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

−26.893387241802933927824607933677, −26.154202665461835547518745572738, −24.95001483762376472009524187995, −23.86026137944614825518299109009, −23.03887139898776894067407523067, −22.09151695650204868240416172634, −20.77816821765938288247068584417, −19.98472732080795323339345312699, −19.099505911398791470462715260969, −17.807323454045777992741144285314, −17.28752940679333018162346330606, −16.13158534482621826414102416669, −15.37093867146826637331000760767, −14.80286602423497206241192987089, −12.558904739714724023027667545660, −11.660575732781736151747941137536, −10.90912572822601227554572320838, −9.85941404235875129270229151930, −8.990867802582391827893018107578, −7.5147655155761606816603957310, −6.85655778239240007379797917471, −5.33277265052464434920980602977, −4.16847921704213178292766338900, −2.55871010212360792456094327968, −0.39688219464733307271788838615, 1.123988399405089639395507691361, 2.66445010497891189492234665279, 4.33690489132509237325667447991, 6.03815648970718783457604673596, 6.96946042741327014045678634992, 8.00863340289909208583536133460, 8.76149044066242592035230298190, 10.4519427623408913494923209949, 11.16132146635503479068045185664, 12.109129585745086015018375224054, 12.789019003241800926361933790493, 14.52551671683540107264753941253, 15.811344433054128442500567212302, 16.67920925737850476974760270149, 17.274609333639205927915794526906, 18.59261485336986517292619372362, 19.17915261677165857880655828855, 19.80904556613362747041365024632, 21.21693767130576898179607725373, 22.23029424917869427100761471742, 23.48919441548647417637021556857, 24.26476068885062025950884121250, 24.86622272375470505800740699445, 26.31935963215060897438027764904, 27.10194546142230099229950528069

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