Properties

Label 1-287-287.118-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: $\chi_{287} (118, \cdot )$
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

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

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