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Properties

Label 1-1287-1287.1069-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.560 - 0.827i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.978 − 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.104 + 0.994i)28-s + (−0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.978 − 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.104 + 0.994i)28-s + (−0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.560 - 0.827i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.560 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.530438538 - 1.342409402i\)
\(L(\frac12)\) \(\approx\) \(2.530438538 - 1.342409402i\)
\(L(1)\) \(\approx\) \(1.640089574 + 0.1856365330i\)
\(L(1)\) \(\approx\) \(1.640089574 + 0.1856365330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−21.12791598216448390839787198530, −20.15764229118353879685832907979, −19.63794253591663464236140981126, −18.73006247096697030869651326287, −18.00159951558520603773127980005, −17.41592215772166014037273500385, −15.919366318127101527395711826, −15.46294675514376108716692706451, −14.42630314674582123544534784814, −14.18961988673060601709595596125, −13.33544863910317208478560692952, −12.33377937058606624256045080574, −11.41026550920191455941029922117, −11.10615375724614312357717905247, −10.25713020597863305546800984815, −9.46106943173795333365432945343, −8.19001871056873098892133113734, −7.27966530819025461616401075691, −6.50716335527625438851573543116, −5.56862178255399302029028009587, −4.716114681418747804527092206, −3.81646209717978639362270247670, −3.03349533497537719549580540084, −2.03359239084277561083173254011, −1.22023099254681415638312742167, 0.391475035019827382962363789031, 1.77223483999094335420819017670, 2.68026639039891464882183485511, 4.045338080592129350414638081779, 4.608112303636359726087526952016, 5.319623102905641859321509562364, 6.11135147829463842915456536793, 7.23717392556694817355945193848, 8.002635695378321016401766689271, 8.63396501837275203246518862686, 9.44697516598157153609377405302, 10.83961175333885932216446782822, 11.76439691738617255290596642401, 12.10250723782076263540667569328, 13.161494782479585893107492693772, 13.74746257080588887853272565378, 14.494774094868641869116876373821, 15.46714818557147379454455601033, 15.91313262932442567417037286195, 16.76485233606327809594211122443, 17.57421344830986094667086526943, 18.01378829215240826810948367298, 19.334618824839099690100136303097, 20.36829626864115305652500166991, 20.747979057440958029951445734636

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