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Properties

Label 1-203-203.101-r0-0-0
Degree $1$
Conductor $203$
Sign $0.309 - 0.950i$
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  + (0.294 − 0.955i)2-s + (−0.997 + 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (0.988 − 0.149i)9-s + (0.563 − 0.826i)10-s + (−0.149 + 0.988i)11-s + (0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (0.365 + 0.930i)16-s + (0.866 − 0.5i)17-s + (0.149 − 0.988i)18-s + (0.997 + 0.0747i)19-s + ⋯
L(s)  = 1  + (0.294 − 0.955i)2-s + (−0.997 + 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (0.988 − 0.149i)9-s + (0.563 − 0.826i)10-s + (−0.149 + 0.988i)11-s + (0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (0.365 + 0.930i)16-s + (0.866 − 0.5i)17-s + (0.149 − 0.988i)18-s + (0.997 + 0.0747i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.309 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8860426905 - 0.6433299893i\)
\(L(\frac12)\) \(\approx\) \(0.8860426905 - 0.6433299893i\)
\(L(1)\) \(\approx\) \(0.9126309226 - 0.4486671038i\)
\(L(1)\) \(\approx\) \(0.9126309226 - 0.4486671038i\)

Euler product

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

−26.91087827350409162803760906070, −25.93626410434514016877600470943, −25.02435026322906924166004762337, −23.97449478657769386706109081496, −23.62949303396185234697089422050, −22.2663947896357865967325227150, −21.69890554364206862250062559248, −20.916111225891949376194136801908, −18.89867768405550186320255887282, −18.13423417463576698234583110421, −17.26047673745459744316444512886, −16.42127558970259076714934617266, −15.87781194660637334388403929208, −14.24019591973214714649772433077, −13.55697465150933451310777328607, −12.57729571779169172837518917231, −11.46586249666040629799425612084, −10.092982675517937415798098691249, −9.07892697307405379825321276040, −7.792460437829404510008100836137, −6.46073915164425976198186891887, −5.80590075487124185485449792823, −4.98810574218525612591040164301, −3.55254231948395088355094218569, −1.28716773189830688936499447708, 1.13178216571005819473516359893, 2.45566791199115964768749848804, 3.96902810420103511008507490415, 5.331033593918507696851337783781, 5.86229391880573165107150195146, 7.38479182269681890171151486335, 9.29835075655936570128601253059, 10.13816064449472720716865529331, 10.76817017437234765683927846791, 11.99998585987524717873616435367, 12.75333281087833381571964569262, 13.755697816014184486135283766585, 14.83832571054560346657150370802, 16.13637471244938548337108772663, 17.51737267356949715061639676200, 18.040213809379871704676923376687, 18.77924552548836541170073227494, 20.45320408654782234480222875643, 20.896445659237363504985575595772, 22.157237629410387751277952538668, 22.56556951701572527201138722335, 23.4208372696059985032720109673, 24.60089012498623569122950968819, 25.78550572183642508915622592922, 26.96948041287722290474360713457

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