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Properties

Label 1-287-287.90-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.999 + 0.0438i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (−0.309 − 0.951i)2-s − i·3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s − 9-s + (0.809 + 0.587i)10-s + (0.587 − 0.809i)11-s + (0.587 + 0.809i)12-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)15-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.309 + 0.951i)18-s + (0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s − i·3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.951 + 0.309i)6-s + (0.809 + 0.587i)8-s − 9-s + (0.809 + 0.587i)10-s + (0.587 − 0.809i)11-s + (0.587 + 0.809i)12-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)15-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.309 + 0.951i)18-s + (0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.999 + 0.0438i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.999 + 0.0438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02295297321 - 1.045898751i\)
\(L(\frac12)\) \(\approx\) \(0.02295297321 - 1.045898751i\)
\(L(1)\) \(\approx\) \(0.5525884645 - 0.5560185803i\)
\(L(1)\) \(\approx\) \(0.5525884645 - 0.5560185803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 - iT \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.587 - 0.809i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + T \)
79 \( 1 + iT \)
83 \( 1 - T \)
89 \( 1 + (-0.951 - 0.309i)T \)
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

−25.97182806399437430963132844613, −24.9176066577426302522247447480, −24.00795260292942238063583117838, −22.99045282681769651285805366912, −22.56862561686788398775412159176, −21.22578367878443998975073954969, −20.247078929774138487094956466421, −19.471266299802289346124156211295, −18.3113123301463161333429111800, −17.12575153953956326194966242495, −16.502616946982709152261511233763, −15.725484980402264233673899298493, −14.98484495890253748025662715140, −14.16030441912315819169609827650, −12.82743305890917189265597067320, −11.60712020150193162803658379054, −10.49762438696071858362392579589, −9.40512581958340561176739124333, −8.71049844190065720021958050449, −7.78727951619231959825734070100, −6.51395160139535426638852318716, −5.301672525555641088115954671635, −4.39897820207814590757306329706, −3.59364671355166838062447904112, −1.15179971609444736528164174324, 0.448364271325802656199966638920, 1.46238446641417878198455451875, 3.0745533111613536678742237545, 3.562591795999540078278639489232, 5.37379255545810751779551217878, 6.75801627004337091040407954090, 7.83820109290798853390476536882, 8.493465188233098504390051721256, 9.77237107667442688172696873826, 11.24442547858006439772143482731, 11.501556284004018514192591618989, 12.46416485566113998102919861953, 13.69483044714681082153088722647, 14.13182054894095547658942169472, 15.71953603000438773612387242984, 16.88798561642359261054914459564, 17.9445085296429348636918531551, 18.72488323793681647170738588518, 19.1999762164195119166555668533, 20.102081777144071225520637466316, 20.98985968438650413026731500227, 22.36841008867723833727170182891, 22.88479299691561626735184182263, 23.726809153199004075353470374611, 24.9087005782634978939976480810

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