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Properties

Label 1-203-203.124-r0-0-0
Degree $1$
Conductor $203$
Sign $0.970 - 0.242i$
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  + (0.149 − 0.988i)2-s + (−0.680 + 0.733i)3-s + (−0.955 − 0.294i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (−0.433 + 0.900i)8-s + (−0.0747 − 0.997i)9-s + (−0.294 + 0.955i)10-s + (0.997 + 0.0747i)11-s + (0.866 − 0.5i)12-s + (−0.900 + 0.433i)13-s + (0.781 − 0.623i)15-s + (0.826 + 0.563i)16-s + (0.866 + 0.5i)17-s + (−0.997 − 0.0747i)18-s + (0.680 + 0.733i)19-s + ⋯
L(s)  = 1  + (0.149 − 0.988i)2-s + (−0.680 + 0.733i)3-s + (−0.955 − 0.294i)4-s + (−0.988 − 0.149i)5-s + (0.623 + 0.781i)6-s + (−0.433 + 0.900i)8-s + (−0.0747 − 0.997i)9-s + (−0.294 + 0.955i)10-s + (0.997 + 0.0747i)11-s + (0.866 − 0.5i)12-s + (−0.900 + 0.433i)13-s + (0.781 − 0.623i)15-s + (0.826 + 0.563i)16-s + (0.866 + 0.5i)17-s + (−0.997 − 0.0747i)18-s + (0.680 + 0.733i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7228131640 - 0.08892410380i\)
\(L(\frac12)\) \(\approx\) \(0.7228131640 - 0.08892410380i\)
\(L(1)\) \(\approx\) \(0.7288399398 - 0.1643947109i\)
\(L(1)\) \(\approx\) \(0.7288399398 - 0.1643947109i\)

Euler product

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

−27.16014093578746986853983117621, −25.68767071585793101790920175549, −24.72416175821429246582340301749, −24.143561804705772380397549105694, −23.14288816033941671744695678158, −22.6084353579751779627938119071, −21.7247500017581924230678215231, −19.820666559953050595265064595855, −19.10697254543549933310709543169, −18.079187328029578869049669267995, −17.17032731722750215916270925337, −16.39583428304634966906271352503, −15.36471954188099563108263299417, −14.36911583007790744225896042191, −13.31779643318349016500086793390, −12.12807510259676711410846708559, −11.60319391314705018797670353824, −9.934478909274150573165975272, −8.53282311846016136657092438605, −7.39443757154107153672926565710, −6.98502810307647691074199336384, −5.605216957900049372887920385345, −4.64186136601108138538072157329, −3.20590980793513257512509956863, −0.79184768345373615778030773248, 1.084059471777784008845096977761, 3.14325684257228449537224625411, 4.15689205852396807518555698871, 4.88732599115075688123417777297, 6.316558338240938281568161762725, 8.00474554287024741393659849162, 9.30419193829653468533887418578, 10.08645940913191436613292058423, 11.27901764116724935155724976669, 11.93942260594438474943388779531, 12.58923256777664581637794760175, 14.40245039092608791887678322904, 14.96018794968714313322147574372, 16.454141989866476804822032920272, 17.085558026596575844522616792377, 18.42004018642111666793807204859, 19.38118415202136423289330364363, 20.21950020999573961816505536386, 21.14851194266541679250650839092, 22.11877652275683770804352768253, 22.81360448596823057862270800235, 23.55385844449869606624839264846, 24.65562346388464604115040998193, 26.57425682993133241325453891584, 26.97312101978211550721763246615

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