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Properties

Label 1-287-287.69-r0-0-0
Degree $1$
Conductor $287$
Sign $0.997 + 0.0733i$
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  + (0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (0.156 − 0.987i)6-s + (−0.951 + 0.309i)8-s + i·9-s + (−0.309 − 0.951i)10-s + (0.453 − 0.891i)11-s + (0.891 − 0.453i)12-s + (0.987 + 0.156i)13-s + (0.453 + 0.891i)15-s + (−0.809 − 0.587i)16-s + (−0.891 − 0.453i)17-s + (−0.809 + 0.587i)18-s + (0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (0.156 − 0.987i)6-s + (−0.951 + 0.309i)8-s + i·9-s + (−0.309 − 0.951i)10-s + (0.453 − 0.891i)11-s + (0.891 − 0.453i)12-s + (0.987 + 0.156i)13-s + (0.453 + 0.891i)15-s + (−0.809 − 0.587i)16-s + (−0.891 − 0.453i)17-s + (−0.809 + 0.587i)18-s + (0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.997 + 0.0733i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.997 + 0.0733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066987702 + 0.03920058505i\)
\(L(\frac12)\) \(\approx\) \(1.066987702 + 0.03920058505i\)
\(L(1)\) \(\approx\) \(0.9787796991 + 0.1487942541i\)
\(L(1)\) \(\approx\) \(0.9787796991 + 0.1487942541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.453 - 0.891i)T \)
13 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (-0.891 - 0.453i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (-0.891 + 0.453i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.156 + 0.987i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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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

−25.606320857503182111399843426886, −24.07332586145253333632145929960, −23.44878166391824083551334030623, −22.52341479452490035627569826106, −22.266704074642750443937523860, −20.92491865971343340107088865989, −20.3185042977547905497498759068, −19.38185863695365859344340233886, −18.29564887936619376569825657, −17.45496935709099631929224504865, −15.96268034476399909890867179257, −15.3744370693342946852827355914, −14.58277370935771273690123098313, −13.24098248431572548011232213498, −12.15076049091207918685760162583, −11.468180651620394237921976209383, −10.770893660515991464133014784452, −9.78811268952042606404967323040, −8.72344017709679326925344999510, −7.00323473057703912199356021197, −5.970685286052009803318386953767, −4.704781576451622171906515105165, −4.011889850012613970906206130444, −3.04242608741157265647848764809, −1.15341411806723722854666015204, 0.83443524784697029024207637341, 3.00022300410534679929726531975, 4.24737530684523735742218175333, 5.23217353030456480817815268000, 6.35886568228822677388493579030, 7.09462158733052990810310387194, 8.186776098637547323495802242204, 8.912621048004284065947936973434, 11.08705455870000499272081529668, 11.61270793781902458341747498714, 12.60254647140447598528937530538, 13.47109553979928148913434189479, 14.29666701982004945191953554128, 15.8984688508052716081927632326, 16.039658310203304259363214341047, 17.14699469343839032326349316652, 18.099588143662397845198156470187, 18.97913813547228269489622191837, 20.067372052026535681883383212665, 21.30435296079143454831938630594, 22.34355048098837575867214401281, 23.06867380161736733065830995366, 23.67344951425426811201805196044, 24.64123147597956695607560387211, 24.94131722763017610631856817903

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