Properties

Label 1-287-287.58-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.848 - 0.529i$
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  + (0.743 + 0.669i)2-s + (−0.258 − 0.965i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.913 − 0.406i)10-s + (0.358 − 0.933i)11-s + (0.933 − 0.358i)12-s + (−0.891 − 0.453i)13-s + (−0.987 − 0.156i)15-s + (−0.978 + 0.207i)16-s + (−0.933 − 0.358i)17-s + (−0.978 − 0.207i)18-s + (0.838 + 0.544i)19-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.258 − 0.965i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.913 − 0.406i)10-s + (0.358 − 0.933i)11-s + (0.933 − 0.358i)12-s + (−0.891 − 0.453i)13-s + (−0.987 − 0.156i)15-s + (−0.978 + 0.207i)16-s + (−0.933 − 0.358i)17-s + (−0.978 − 0.207i)18-s + (0.838 + 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2644331854 - 0.9239752340i\)
\(L(\frac12)\) \(\approx\) \(0.2644331854 - 0.9239752340i\)
\(L(1)\) \(\approx\) \(1.141621420 - 0.1667868386i\)
\(L(1)\) \(\approx\) \(1.141621420 - 0.1667868386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 + (0.358 - 0.933i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (-0.933 - 0.358i)T \)
19 \( 1 + (0.838 + 0.544i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.998 + 0.0523i)T \)
53 \( 1 + (-0.777 - 0.629i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.207 - 0.978i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.544 - 0.838i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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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

−25.83921643688270421169703465964, −24.62338488909479244693933645814, −23.53460239894462247268472956876, −22.53206659191332311384747376870, −22.06164618464360306181494179610, −21.526768904362291829375480101149, −20.217893486930356206193746627815, −19.81325204741014735726006546170, −18.36149618230020526070109644305, −17.59456801586397319174998717365, −16.31489432433760801176876589742, −15.10643947330064735205046741584, −14.70563799168608957368341245278, −13.7987913102685282947866112400, −12.47591837727063183106977427488, −11.5072041206146795784746543863, −10.72539632027408190901619163406, −9.84172551499660406348920591229, −9.2141283737096281364708244081, −7.07960633111996207688951317885, −6.178970265252877658897153380601, −4.98570011347380204908235855516, −4.16786629423696702224661913385, −2.987312260106720298692421792252, −1.97112954187383714389011238469, 0.210222071102776099345379797004, 1.810123058988875606875497639745, 3.16432042174140142693534977213, 4.730847534907611760381098552123, 5.6280212217500410934723200923, 6.38562006084339310372068250081, 7.62594525250892843890994003205, 8.34380282302645556276185247229, 9.50551179049451262182373048647, 11.390373918656738106698570009926, 12.06756467454402554099604763257, 13.04675225781897995152351842630, 13.633096556871885352653260556994, 14.47860065617265424719750204603, 15.89555660649298628959231028960, 16.67976652275222413394220108658, 17.450659301721275076284017488189, 18.20549743090014618605948862026, 19.67750135847351669159929777497, 20.36441832863460047205086403269, 21.68133720758622492147833206652, 22.28105623857829756380636270813, 23.37503391502475650637172841735, 24.24029591261497566598046571471, 24.711674373014824272357295103379

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