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Properties

Label 1-1287-1287.943-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.923 + 0.383i$
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  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.207 − 0.978i)5-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.951 − 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (−0.743 − 0.669i)28-s + (−0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.207 − 0.978i)5-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (−0.994 + 0.104i)19-s + (−0.951 − 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (−0.743 − 0.669i)28-s + (−0.913 + 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.923 + 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3436664574 - 1.724710437i\)
\(L(\frac12)\) \(\approx\) \(-0.3436664574 - 1.724710437i\)
\(L(1)\) \(\approx\) \(0.9391853189 - 1.067115886i\)
\(L(1)\) \(\approx\) \(0.9391853189 - 1.067115886i\)

Euler product

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

−21.61105805644222312885714098415, −21.03773184708650357275781362112, −20.046989563525019110367347629925, −18.974359973245296320022596504621, −18.19260865320063723970268799880, −17.62939697342627020053791682845, −16.86633821534033961594531684266, −15.730897592768061951996483386231, −15.22765147117879951963848638135, −14.596763794278593605879859865173, −13.95597525462578107253695305261, −13.04261768758124326398763585320, −12.30929954194152278744372040462, −11.26874532044184233921522688666, −10.93266479654496624572001905893, −9.53217540224292512471295670792, −8.6852904995221801526584223842, −7.80980312211110193624757069940, −7.09215809751263456974599938267, −5.98721243609301251887482164774, −5.810136446035616050064996408201, −4.44073466762781845067221344733, −3.83242707191655838078443790155, −2.43563944913370389974932759349, −2.201267671822066454062649005980, 0.48120470943852856203455850724, 1.62718739491025409019205512167, 2.26756177024564535718028892209, 3.76842092138966904995848799291, 4.32372084427260030458032178574, 5.05581523811837698347511549405, 5.93385599148334130332328888352, 6.88218915952715109624408713804, 7.99521253446590203546794323564, 8.92730546122668747424295721048, 9.717007863877485476253318682563, 10.649339742740973374735963114319, 11.24763838677000150900535222554, 12.13818039358201495808901456747, 12.975561331556008629736101809044, 13.41610660112197311279868525130, 14.294462632047512545075348382172, 14.95105880399693007435415786198, 16.01221814358256522435736446772, 16.65198117102365326402081478551, 17.64435419645861683841788045928, 18.26003161504914244050479750235, 19.48756896328200312786368367940, 20.03273717893135057139830573918, 20.544969118896993911677767797896

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