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Properties

Label 1-203-203.159-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.895 + 0.446i$
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  + (0.680 + 0.733i)2-s + (0.563 + 0.826i)3-s + (−0.0747 + 0.997i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (−0.365 + 0.930i)9-s + (−0.997 − 0.0747i)10-s + (0.930 − 0.365i)11-s + (−0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (−0.988 − 0.149i)16-s + (−0.866 − 0.5i)17-s + (−0.930 + 0.365i)18-s + (−0.563 + 0.826i)19-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)2-s + (0.563 + 0.826i)3-s + (−0.0747 + 0.997i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.781 + 0.623i)8-s + (−0.365 + 0.930i)9-s + (−0.997 − 0.0747i)10-s + (0.930 − 0.365i)11-s + (−0.866 + 0.5i)12-s + (0.623 − 0.781i)13-s + (−0.974 − 0.222i)15-s + (−0.988 − 0.149i)16-s + (−0.866 − 0.5i)17-s + (−0.930 + 0.365i)18-s + (−0.563 + 0.826i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.895 + 0.446i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.895 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3851632047 + 1.636423465i\)
\(L(\frac12)\) \(\approx\) \(0.3851632047 + 1.636423465i\)
\(L(1)\) \(\approx\) \(0.9667826560 + 1.153479113i\)
\(L(1)\) \(\approx\) \(0.9667826560 + 1.153479113i\)

Euler product

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

−26.462397140360940128589082766899, −25.17119577895938504287054647321, −24.29359167232198205236439025383, −23.663810154441373182678080858844, −22.855495395680634940902521231354, −21.55300777336615893459143870387, −20.58383104696832236352397135671, −19.686395663219443478819965006888, −19.333655921846645804707362181525, −18.1664242592100659386826272373, −16.8819663699657881381667650837, −15.369300218690982115493702553572, −14.70142682544677323898337781730, −13.41738942585843590094902546493, −12.89513580659441859325769653554, −11.796055422093493297733689155531, −11.17633060670877698398636992238, −9.27462922210421822911967378096, −8.73769575190082810138866509494, −7.150938915192980479984453640890, −6.17045518461326492091468252426, −4.49068001754682266078533860755, −3.71939135665533313695232542327, −2.21469575459680217676853924291, −1.06406931104100865118753359752, 2.781711067693830333294973482, 3.67342072442003704488154657666, 4.512015140467591539613124539117, 5.94156936844810777212784483784, 7.103904546521606920042716351462, 8.239249645206758723509338512903, 9.01727506633964319759285527830, 10.65948148985247098834679253917, 11.488452788751355221964506297946, 12.85193834024088925489060268651, 14.074417025208745525774973702396, 14.70534363534239510238229639774, 15.56976515082876996329767132138, 16.2475242551405218727824377171, 17.36584306742965408516562628924, 18.70430173832198619602848854492, 19.8580225253745529003217999647, 20.75650240982099366882289541404, 21.835336317680189139709653487211, 22.58218756959510083034940462753, 23.217015184034238806929686480147, 24.60428981643964238868256442985, 25.34468246298815267164773566024, 26.22382592843764252071902230847, 27.260761677398120579268108228061

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