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Properties

Label 1-1287-1287.1220-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.671 + 0.740i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·7-s − i·8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + i·20-s + 23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + i·7-s − i·8-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + i·20-s + 23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3499740668 + 0.7899922434i\)
\(L(\frac12)\) \(\approx\) \(0.3499740668 + 0.7899922434i\)
\(L(1)\) \(\approx\) \(1.291589323 - 0.1204333268i\)
\(L(1)\) \(\approx\) \(1.291589323 - 0.1204333268i\)

Euler product

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

−20.86017317103832369213190212374, −19.83787740717448755775306933620, −19.34974092141348836890706045384, −18.08838677053433933698717105451, −17.06856819349259856447280628893, −16.66110668471318175768956580432, −15.89306201831114326436515590610, −15.195286735209220646531201584470, −14.359196115551226623965253884100, −13.64982493603894282024381500561, −12.78889599361537478741905098006, −12.29405097788832048529557842749, −11.195412606390518514213202334094, −10.81644365178075360549083326013, −9.34995395152734565692376306157, −8.48888693557570905907010768198, −7.52766147568433161559739634171, −7.173786563017882402964326754932, −6.15570443494523066271259835065, −4.91607863005563990886428024962, −4.54564423894809812887190429610, −3.58718599580143330579321303462, −2.83945661059591031444841366577, −1.295012448990950717475641409667, −0.12323299367721207748367128994, 1.30974101344029717349012606767, 2.45055796460469237089919115031, 3.12558304923761274320320810540, 4.06608433051137910453279847397, 4.80795936300015432808209522578, 5.98419665945144071058737894470, 6.400217496320134476311365595850, 7.62573166188974149317343355671, 8.40519583372979793348513694551, 9.53661031520246185400542708491, 10.39580300869535803386151467116, 11.25750876311225329743256664012, 11.75834524976687907358813225572, 12.60172560130874451469615377071, 13.128151612761287034532177187792, 14.39636827190474320033488126799, 14.940584285448249207856438844621, 15.371009483062738364164741553789, 16.23579074265044806000364035053, 17.23664886066886984501081999192, 18.5618708774708938223894098916, 18.945928041621361247866669696480, 19.47280733822683024474575609976, 20.45055063775738966929152537313, 21.28102130074123104589795336665

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