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Properties

Label 1-1287-1287.1051-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.595 - 0.803i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.595 - 0.803i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.595 - 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5321333672 + 1.056373252i\)
\(L(\frac12)\) \(\approx\) \(-0.5321333672 + 1.056373252i\)
\(L(1)\) \(\approx\) \(0.7027331567 + 0.8970299532i\)
\(L(1)\) \(\approx\) \(0.7027331567 + 0.8970299532i\)

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

−20.516816914656060900353085178365, −20.046771683365737595548210927172, −19.12231777101339511564738242326, −18.52315020878474526227122103522, −17.39640019634863279363070093443, −16.71113405574702709020300974844, −15.934930220224380396672283481921, −15.00910860213197416199887537177, −14.00470934518414658381418254524, −13.437856047420561370014893527459, −12.83376184485699214190819695123, −12.13057424863685730029357169607, −11.279244461808662259342160838117, −10.22888199164936912702871578936, −9.7799555002497192978752435632, −8.97153419404233048021523391005, −7.99681519527098967208357727957, −6.599935824707401845612211224886, −6.07072208632595853634648876113, −4.8737818026159852927849476878, −4.42271722000119682517683663281, −3.36087652102146236178624847620, −2.44257454020275541285038669571, −1.35711152354427149102732634882, −0.36714656444081331177337734218, 1.977880595819632553295571781554, 2.97218147804621320278887954634, 3.5598151952956291258086042816, 4.74445717441361281767961487250, 5.73713546243816491433785608158, 6.36093790845911961950283337983, 6.93232390111037951740445893721, 7.92567984600084790576660626390, 8.8928418763782365996661794569, 9.57868813218113558341579321137, 10.64803319052116789423346421356, 11.50759980724140423051980666507, 12.505930219591234852133059450573, 13.14723414636759642009196049523, 13.81235222057893048837440458817, 14.89128779866828722806507205105, 15.14007507523069495297495485567, 15.95762713355054456435801594223, 16.93286578358816430003996330654, 17.546326210407652712677741921455, 18.35518185117086018978604923461, 19.076750164014025973828107864272, 19.88609845167569948778454002653, 21.28107530584364607186073915134, 21.657781890874370640916365848287

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