Properties

Label 1-287-287.261-r1-0-0
Degree $1$
Conductor $287$
Sign $0.959 + 0.280i$
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  + (−0.207 − 0.978i)2-s + (0.258 + 0.965i)3-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (0.891 − 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.104 − 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.629 − 0.777i)12-s + (−0.453 − 0.891i)13-s + (0.156 + 0.987i)15-s + (0.669 − 0.743i)16-s + (0.629 − 0.777i)17-s + (0.669 + 0.743i)18-s + (0.998 − 0.0523i)19-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.258 + 0.965i)3-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (0.891 − 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.104 − 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.629 − 0.777i)12-s + (−0.453 − 0.891i)13-s + (0.156 + 0.987i)15-s + (0.669 − 0.743i)16-s + (0.629 − 0.777i)17-s + (0.669 + 0.743i)18-s + (0.998 − 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.959 + 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.155988022 + 0.3090956222i\)
\(L(\frac12)\) \(\approx\) \(2.155988022 + 0.3090956222i\)
\(L(1)\) \(\approx\) \(1.245762738 - 0.04923098246i\)
\(L(1)\) \(\approx\) \(1.245762738 - 0.04923098246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.994 + 0.104i)T \)
11 \( 1 + (0.777 + 0.629i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (0.629 - 0.777i)T \)
19 \( 1 + (0.998 - 0.0523i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.838 - 0.544i)T \)
53 \( 1 + (-0.358 + 0.933i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.743 + 0.669i)T \)
67 \( 1 + (0.933 + 0.358i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.0523 - 0.998i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−25.10472568536318936105258646670, −24.47025268849787872324942828797, −23.92163975808050714333682846682, −22.71940883373485646808595239317, −21.89594578946770423010322270773, −20.74030594565424628331525960157, −19.28294887791005808450359732802, −18.894340971932624898152306997882, −17.763429676646146038077899903273, −17.098329585900952005813932069950, −16.40962148053438862714204112739, −14.754329737187375883383290134311, −14.236712643260043628358656919228, −13.43951777300888385683859681023, −12.574059164692015671540803250899, −11.22943974559303758832679562099, −9.58679260600222757294939043902, −9.09948230175515583591623941646, −7.939956046463562522065037090268, −6.93412454701379493077658126495, −6.121162105396482431487369830426, −5.29778907841525539381017897589, −3.61376558681222550764823268459, −1.89864090122889748180752483068, −0.840605437083785533154633764016, 1.1538305827483653974821897979, 2.597177664450477480766309886466, 3.37342891824295020371728937431, 4.79777861346439938189826595573, 5.476465392359809598063750881552, 7.32020047349322195721676094527, 8.73586164182141367105416556363, 9.597456926406550201168113361321, 10.04159011726988131286519952640, 11.06515028639240283629693160259, 12.130484916013211452529843950349, 13.26276081821148858327285972327, 14.21719253574364275362367897127, 14.90205154616421674089129402709, 16.41626821368289452914492685231, 17.27575085618862779652623798062, 17.979461909543340483698696068630, 19.167329113145656594533380804824, 20.311612701348967804070163363, 20.63140229371729320037930888923, 21.66324073510641342598254447856, 22.41821913894236273905527723218, 22.877035191340525692455314542623, 24.84133461891698489347244065311, 25.49564315523111731543587991278

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