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Properties

Label 1-1287-1287.635-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.472 + 0.881i$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.743 − 0.669i)5-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (0.406 − 0.913i)19-s + (0.951 + 0.309i)20-s + 23-s + (0.104 + 0.994i)25-s + (−0.207 + 0.978i)28-s + (−0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.743 − 0.669i)5-s + (0.587 − 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.669 + 0.743i)17-s + (0.406 − 0.913i)19-s + (0.951 + 0.309i)20-s + 23-s + (0.104 + 0.994i)25-s + (−0.207 + 0.978i)28-s + (−0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.472 + 0.881i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.472 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3829929106 - 0.6398886923i\)
\(L(\frac12)\) \(\approx\) \(-0.3829929106 - 0.6398886923i\)
\(L(1)\) \(\approx\) \(0.5683052853 - 0.5177706496i\)
\(L(1)\) \(\approx\) \(0.5683052853 - 0.5177706496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.406 - 0.913i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.207 - 0.978i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.406 + 0.913i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−21.49656364287478911393028081235, −20.55346927731371081860738474775, −19.49396294404532081284451320068, −18.78043244374140850760788439939, −18.31215449711836879282956399762, −17.57938853724094305525484873497, −16.67917892064941896530591957030, −15.66799941918352884373044793039, −15.44133796335486422297555068020, −14.52009271865083154242348041757, −14.02928638518302386221638432759, −12.93631386345352403555704639518, −11.95570686321217578534789258784, −11.24869426180550149495244845800, −10.31588373244440816206223703198, −9.35520780248130618112486014254, −8.53113587898648001002597915753, −7.87956232598259645742879530849, −7.086163081341540407704965489889, −6.33643787490311843514044786180, −5.338167420227114072184845081051, −4.62295728426252772154106858729, −3.5962351877563285354875877311, −2.517522738557390749750610127044, −1.136341581218282175848385685724, 0.20363909521880983886933391808, 0.99288667397919990157470843196, 1.925089085422644713045357922954, 3.17260229931685111003711534983, 4.080999893288878760241547328861, 4.612494090713663444615148078575, 5.49829828777615113153617697813, 7.17351137794120456051535112402, 7.67877382585088159679491099717, 8.82438949134687636922587204377, 9.05186827372051370804420262582, 10.42555894515484957004941168890, 10.9869683076628166243369212827, 11.597119664649327366769252730373, 12.54323096996749467644980664998, 13.14073870829588320981016372531, 13.89817997505707114664706957267, 14.85920403777695018250670028723, 15.747874551252361761062658376675, 16.82538436207327517512380605383, 17.26772325270560901496837535443, 18.047294012080283820921524916098, 19.0750365696565282478122419932, 19.63241098989040558218117293496, 20.37261318792874618522251265266

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