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Properties

Label 1-287-287.243-r0-0-0
Degree $1$
Conductor $287$
Sign $0.999 + 0.0239i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.866 + 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + i·8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)10-s + (0.965 − 0.258i)11-s + (0.258 − 0.965i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.5 + 0.866i)18-s + (0.965 + 0.258i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.707 + 0.707i)6-s + i·8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)10-s + (0.965 − 0.258i)11-s + (0.258 − 0.965i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.5 + 0.866i)18-s + (0.965 + 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.999 + 0.0239i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.999 + 0.0239i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056355905 + 0.01264706835i\)
\(L(\frac12)\) \(\approx\) \(1.056355905 + 0.01264706835i\)
\(L(1)\) \(\approx\) \(0.9269855476 + 0.07183906191i\)
\(L(1)\) \(\approx\) \(0.9269855476 + 0.07183906191i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-0.258 - 0.965i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.965 + 0.258i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.258 + 0.965i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−25.77645576292170561413426902541, −24.64661340579033447844496579304, −24.24418613408790355478509754181, −22.53146042887531224821781047124, −21.63242612638203120224647998948, −20.66526169769609852670692576149, −19.896239311631422436166991006791, −19.4122408867111228337700159422, −18.61413994370711060619237579536, −17.18700132431354847003193352523, −16.48056387740447677272745975395, −15.474394292381802397477373417891, −14.64309510584686012835156280147, −13.28647420322073434781344343900, −12.2573949506966844446482262832, −11.50413316165456518344433441292, −10.22395485805413909514409232439, −9.2557183155064256678434573958, −8.62904423690310143192058112567, −7.66171276583454482509876278293, −6.79944402516344513257492464765, −4.503267055208929626167217010702, −3.80258487414514599566003337944, −2.55061903177885329305784100629, −1.28150235794123564513494211176, 1.03772302949655896943910656843, 2.606637209903385856051526655808, 3.64176703796404535123505502065, 5.27007122822011237928033869259, 7.014934520715009533209781397072, 7.23233250407810232389340467971, 8.38582205564748475055093727385, 9.20761847311511468068127075789, 10.18132345440149250533728326750, 11.386054586277419180292266484, 12.32379972489618732372623025579, 13.98940196231156642084004192002, 14.51358940851203702191676898765, 15.51082309852174880361487745150, 16.113121117394313922605893060483, 17.50999569248069765084382883872, 18.340844343452770478709184355022, 19.23784812468925119135328339990, 19.83413680760397100100718529135, 20.44880382196501408892002941950, 21.9938867701510524977961424554, 23.07734386474888033329351063279, 24.05708849415839508281717131645, 24.91680189093182124605524254205, 25.40068720339788631855109127359

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