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Properties

Label 1-203-203.100-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.451 - 0.892i$
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  + (−0.365 + 0.930i)2-s + (−0.955 − 0.294i)3-s + (−0.733 − 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)8-s + (0.826 + 0.563i)9-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.5 + 0.866i)12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)15-s + (0.0747 + 0.997i)16-s + (0.5 − 0.866i)17-s + (−0.826 + 0.563i)18-s + (−0.955 + 0.294i)19-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)2-s + (−0.955 − 0.294i)3-s + (−0.733 − 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.900 − 0.433i)8-s + (0.826 + 0.563i)9-s + (0.733 + 0.680i)10-s + (−0.826 + 0.563i)11-s + (0.5 + 0.866i)12-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)15-s + (0.0747 + 0.997i)16-s + (0.5 − 0.866i)17-s + (−0.826 + 0.563i)18-s + (−0.955 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1338391842 - 0.2177153642i\)
\(L(\frac12)\) \(\approx\) \(0.1338391842 - 0.2177153642i\)
\(L(1)\) \(\approx\) \(0.4912252057 + 0.005805338578i\)
\(L(1)\) \(\approx\) \(0.4912252057 + 0.005805338578i\)

Euler product

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

−27.20459606299623121380075777764, −26.48712081753391491380823356474, −25.734275519974702206766293592255, −23.96546876802934319563661990823, −23.16537932793380896248118306833, −21.96426570010927401387963453755, −21.74535699610319195484413395610, −20.76739358323894730206589530552, −19.18540799305422817962832898609, −18.711493060651761866980821725827, −17.55523238852346781501962316925, −17.05765175030506594117355674138, −15.73264115476050939704682146543, −14.442225050236904864180211066429, −13.25829188835605735061063102613, −12.18879347293406597954131266343, −11.280357365623200203642737586096, −10.32524429243610905649696522383, −9.92861555316201790757701504328, −8.32403071217463273056947324683, −6.98271737080667852154940134880, −5.739850680465613586005542576831, −4.47381063954893834121479280277, −3.183204165824244760363684658454, −1.86142287626480284817515324693, 0.24038008041894646152197185204, 1.861401531460725683262707030749, 4.60197369553617076698873443426, 5.19659864180727088016964929019, 6.19851332126860964464791305442, 7.41021543137822629301948201231, 8.27821775306285516808916738853, 9.77457612070695872915704578838, 10.33149752306221642079336880923, 12.06489627063461940940260619832, 12.86633637173251907419910186465, 13.84813129088686614166024552125, 15.263506797070130263275323885, 16.18517485027554476175491352195, 16.96119174321996791208552834731, 17.68899857505918277664111551413, 18.47722423095044741762177224191, 19.656332543852775049046193173109, 20.93078292438476247637295428736, 22.1103320092726792906775133916, 23.100276966914816612437276700633, 23.780354654092662422289243370767, 24.7011123026329327727369934413, 25.28094695978517840686490561026, 26.54198562983850854474103509972

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