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Properties

Label 1-1287-1287.1172-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.338 - 0.940i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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 + (−0.309 + 0.951i)4-s + (0.406 + 0.913i)5-s + (0.743 − 0.669i)7-s + (−0.951 + 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.104 − 0.994i)17-s + (0.743 + 0.669i)19-s + (−0.994 + 0.104i)20-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (0.406 + 0.913i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.406 + 0.913i)5-s + (0.743 − 0.669i)7-s + (−0.951 + 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.104 − 0.994i)17-s + (0.743 + 0.669i)19-s + (−0.994 + 0.104i)20-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (0.406 + 0.913i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.338 - 0.940i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.338 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4844729699 - 0.3406068492i\)
\(L(\frac12)\) \(\approx\) \(0.4844729699 - 0.3406068492i\)
\(L(1)\) \(\approx\) \(1.115368296 + 0.6279699576i\)
\(L(1)\) \(\approx\) \(1.115368296 + 0.6279699576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.743 - 0.669i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (0.743 + 0.669i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.994 + 0.104i)T \)
37 \( 1 + (-0.743 + 0.669i)T \)
41 \( 1 + (-0.743 - 0.669i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.994 - 0.104i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−21.11203262909420246824777058528, −20.16377850818675926444870469622, −19.82619995504580950787922882754, −18.749410572737028362001219995642, −17.96058019158898466295441778898, −17.416332066684060329606111213174, −16.2680431334202807734940957439, −15.4307127559734647947361549439, −14.6949852033080032515561410581, −13.85361476716477935132205849829, −13.19434309876482570001777430557, −12.345331503095402425885428454467, −11.861946471308472637638002970, −10.96386469892277600959683579204, −10.128940166531873740683950122187, −9.13882701147931380400001341713, −8.73687334535535047881437715628, −7.6015538245456422281718813497, −6.208004187220246068326724360635, −5.431410769367872671205707957750, −4.94537077535763583143842728388, −3.97460627884174671719342203475, −2.94537023014858127932805843343, −1.745795429761824340197041688236, −1.39940159778730074386593931454, 0.08286125905475079957726622016, 1.70371772542305528331756617587, 2.855164760044122626385846769267, 3.6614082315749481799968736224, 4.63785081230755075951751536805, 5.44507523985822510653026428387, 6.33750404717930813052148475861, 7.15167771355425346466086845060, 7.69743325025632986409621261160, 8.60126116454720270163164447005, 9.7436115537826861572799642094, 10.48705664361144026037415327051, 11.54315166630250333168594135329, 12.07251794168809303142827484757, 13.44589846535764566543785666635, 13.77401331666537851725435908622, 14.55644230056158631141194486390, 15.02878193492953494972898290167, 16.13125526395290700342862765132, 16.714042710708997549777591364645, 17.674790418127700615304485964640, 18.11882466683862577093880770966, 18.84444856572381067535100675844, 20.225967773747763833633018004447, 20.79818379356340223560903910392

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