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Properties

Label 1-1287-1287.1094-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.209 + 0.977i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.994 − 0.104i)5-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.743 − 0.669i)19-s + (0.587 + 0.809i)20-s + 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.994 − 0.104i)5-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.743 − 0.669i)19-s + (0.587 + 0.809i)20-s + 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.209 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7669608461 + 0.6199708175i\)
\(L(\frac12)\) \(\approx\) \(0.7669608461 + 0.6199708175i\)
\(L(1)\) \(\approx\) \(0.7167766143 + 0.3214717087i\)
\(L(1)\) \(\approx\) \(0.7167766143 + 0.3214717087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
5 \( 1 + (-0.994 - 0.104i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.406 - 0.913i)T \)
37 \( 1 + (-0.743 + 0.669i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.743 - 0.669i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.994 - 0.104i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
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

−20.86239945852314830204045312771, −19.97428194601183331735310606156, −19.46912703841978438442134514089, −18.57332146171577596277293328818, −18.06248158287343555494295118104, −17.12326867783110288806695716898, −16.476977241825699407452103929400, −15.495705922048928628561520054029, −14.579453038060433318169960457430, −13.8619333686792231204904764124, −12.92980909361055111912169368492, −11.873647226726224239569741843343, −11.715822273199424596981640565414, −10.62611530280264845227329933323, −10.27269045887993572793356312800, −8.757683970089630344357227984726, −8.54144561559346360476354683812, −7.485729025054348328177482555979, −6.92045871957164950114240055846, −5.14235022815998793745650487344, −4.559746301413499052086330535256, −3.66802252799599046513536192627, −2.82689837785489880327384742973, −1.69162276923858311873406781267, −0.664228421648095020827464585612, 0.837635095388767073697547232701, 1.98283882578466111647324496587, 3.46786301691259667321208017989, 4.62952193113766575299251691272, 4.9109659549402217641057616206, 6.20535233076472601903903793618, 6.9300130882999885699978042442, 7.928436918264516892124378342619, 8.38095348288802412526319553259, 9.00947651739270162935547088812, 10.23917684246497455697383195242, 10.99528074654746198071614609244, 11.74109332175933867397958117913, 12.76991945399226529716051352046, 13.60676978059278506464685157585, 14.66466359855617976704948698739, 15.181663951352474521070307913618, 15.587058585227716842468654972786, 16.73094306485383884796509109684, 17.226766950357249346276142498462, 18.015170833109687317198514999948, 18.94084871552944080260777295843, 19.34108026198154643739802258254, 20.24173859141048165857178369974, 21.19735405978682232411942893440

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