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Properties

Label 1-2563-2563.9-r0-0-0
Degree $1$
Conductor $2563$
Sign $0.926 + 0.375i$
Analytic cond. $11.9025$
Root an. cond. $11.9025$
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  + (0.183 + 0.983i)2-s + (0.360 − 0.932i)3-s + (−0.932 + 0.360i)4-s + (0.439 − 0.898i)5-s + (0.983 + 0.183i)6-s + (0.755 + 0.655i)7-s + (−0.524 − 0.851i)8-s + (−0.740 − 0.671i)9-s + (0.963 + 0.267i)10-s + i·12-s + (0.971 − 0.236i)13-s + (−0.506 + 0.862i)14-s + (−0.679 − 0.733i)15-s + (0.740 − 0.671i)16-s + (−0.236 + 0.971i)17-s + (0.524 − 0.851i)18-s + ⋯
L(s)  = 1  + (0.183 + 0.983i)2-s + (0.360 − 0.932i)3-s + (−0.932 + 0.360i)4-s + (0.439 − 0.898i)5-s + (0.983 + 0.183i)6-s + (0.755 + 0.655i)7-s + (−0.524 − 0.851i)8-s + (−0.740 − 0.671i)9-s + (0.963 + 0.267i)10-s + i·12-s + (0.971 − 0.236i)13-s + (−0.506 + 0.862i)14-s + (−0.679 − 0.733i)15-s + (0.740 − 0.671i)16-s + (−0.236 + 0.971i)17-s + (0.524 − 0.851i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2563\)    =    \(11 \cdot 233\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(11.9025\)
Root analytic conductor: \(11.9025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2563} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2563,\ (0:\ ),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.378262272 + 0.4635931825i\)
\(L(\frac12)\) \(\approx\) \(2.378262272 + 0.4635931825i\)
\(L(1)\) \(\approx\) \(1.451857001 + 0.2200876206i\)
\(L(1)\) \(\approx\) \(1.451857001 + 0.2200876206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
233 \( 1 \)
good2 \( 1 + (0.183 + 0.983i)T \)
3 \( 1 + (0.360 - 0.932i)T \)
5 \( 1 + (0.439 - 0.898i)T \)
7 \( 1 + (0.755 + 0.655i)T \)
13 \( 1 + (0.971 - 0.236i)T \)
17 \( 1 + (-0.236 + 0.971i)T \)
19 \( 1 + (0.985 - 0.172i)T \)
23 \( 1 + (0.976 - 0.214i)T \)
29 \( 1 + (-0.524 + 0.851i)T \)
31 \( 1 + (0.487 + 0.873i)T \)
37 \( 1 + (-0.118 + 0.992i)T \)
41 \( 1 + (-0.978 - 0.204i)T \)
43 \( 1 + (0.214 + 0.976i)T \)
47 \( 1 + (0.999 - 0.0108i)T \)
53 \( 1 + (-0.193 + 0.981i)T \)
59 \( 1 + (0.978 - 0.204i)T \)
61 \( 1 + (0.622 - 0.782i)T \)
67 \( 1 + (0.827 + 0.561i)T \)
71 \( 1 + (0.940 + 0.339i)T \)
73 \( 1 + (-0.151 - 0.988i)T \)
79 \( 1 + (0.496 - 0.867i)T \)
83 \( 1 + (0.703 - 0.710i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−19.425447683269198586673853993095, −18.747510530171356058507151420451, −18.07384299671454583539348478259, −17.387888658834803843205672049266, −16.637437585334592573081356772716, −15.51915620859373752358632362654, −14.982786933960906392443417887828, −14.0196687786334522370694182868, −13.86733609379041731757439595218, −13.22046452815841443649885555101, −11.725183854688983928083277824536, −11.23751187131526306381251619101, −10.84044486652734306763497985226, −9.9645300528214756623233471954, −9.51676803127255244773048814323, −8.68084732734208937092623840808, −7.818129045266351112420594241266, −6.87394421837246932174956737224, −5.58317142389663336467522895497, −5.16306443643338623496387617102, −4.01246836537784689703537037010, −3.67895253131302307368768209971, −2.69725444022707948938628797215, −2.04246588686208895953715672273, −0.87555786990359910484418591366, 1.02633347682323589482359634222, 1.5523635406598655395876002154, 2.821919438226122588725255889401, 3.74468210399699990538322125261, 4.90330100916896979676655908190, 5.42893588513895386040211066541, 6.16292496525578547008071556527, 6.8772557091522144767903521633, 7.86535498297898672096422148064, 8.48209014494383478535890271712, 8.79878332582756865237608562525, 9.57601382730079690097121946950, 10.87424857101318987070614574322, 11.904027110151873296176626259950, 12.52777209587940177111802836132, 13.15382621832746602459907041193, 13.719225245786058002724420537550, 14.39985118912812476575090553516, 15.16563452696730117154747057031, 15.794187570665629549323060763413, 16.71396275755983666720410728124, 17.43339695395809514963121269032, 17.86364613101093255523578014137, 18.53847942983465034897035491744, 19.16681115470519923618011881347

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