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Properties

Label 1-1287-1287.553-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.340 + 0.940i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.743 + 0.669i)5-s + (0.994 + 0.104i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (0.994 − 0.104i)19-s + (0.207 + 0.978i)20-s + (0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (0.743 + 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.743 + 0.669i)5-s + (0.994 + 0.104i)7-s + (0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (0.994 − 0.104i)19-s + (0.207 + 0.978i)20-s + (0.5 − 0.866i)23-s + (0.104 + 0.994i)25-s + (0.743 + 0.669i)28-s + (−0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.599861993 + 3.929484357i\)
\(L(\frac12)\) \(\approx\) \(5.599861993 + 3.929484357i\)
\(L(1)\) \(\approx\) \(2.526332366 + 0.9989612244i\)
\(L(1)\) \(\approx\) \(2.526332366 + 0.9989612244i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (0.743 + 0.669i)T \)
7 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.207 - 0.978i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.207 - 0.978i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.207 - 0.978i)T \)
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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

−20.711363621266196140953592538090, −20.31747981507470828590848190412, −19.38103175308074215524472575230, −18.341695123145089916557291792428, −17.64907823012411003170138825534, −16.66703336351593780990766926216, −16.11440774911645402715833765951, −15.036273395372241129186742039666, −14.30666009194528580149410877510, −13.80124240572388023435644342560, −12.97323608084679120490678788625, −12.21785491082571737178752603499, −11.48938171835630544605228073745, −10.69260725632505864311302252734, −9.79559580591007235967538406366, −9.04446042867470035228170514368, −7.811658278005516190805746318453, −7.13776908436017955170673339033, −5.80690625405959439879822522171, −5.36811189958064981892425927553, −4.6703892392294671233261063851, −3.635693725278515178300172957045, −2.62238304502357961752372294978, −1.50748332091096988362802271897, −1.05657077933494525061147023865, 1.222220785937229439222668316700, 2.23094589963293099677206302034, 2.97739278093878240479750344341, 4.02203328394840477629378430152, 5.00692358544268804446984825279, 5.70673098973372534910952882733, 6.39673869791179432193211010964, 7.510785621174396021698487371144, 7.909910275190701502345372648830, 9.215471801208008749699250193045, 10.17324295000170301457697778471, 11.15149162013252014953452910054, 11.54427487480890654693733567665, 12.64575944905884072684341010017, 13.369108676685627797185850543966, 14.23829985710145526348317062758, 14.644077973656415566838776610504, 15.25770237597823375354078037141, 16.4219923636940280580009101088, 17.00715329081078884507327986839, 17.896338986878639175222000351485, 18.490909694622784471267320755764, 19.5498671035167469211214857188, 20.77382432684366939280678248294, 20.917301598935321162000580708200

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