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Properties

Label 1-287-287.11-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: $\chi_{287} (11, \cdot )$
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

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

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