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Properties

Label 1-1287-1287.475-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.169 - 0.985i$
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  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.994 − 0.104i)5-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.587 + 0.809i)20-s − 23-s + (0.978 − 0.207i)25-s + (0.406 − 0.913i)28-s + (0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.994 − 0.104i)5-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.587 + 0.809i)20-s − 23-s + (0.978 − 0.207i)25-s + (0.406 − 0.913i)28-s + (0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7338965933 - 0.8712528109i\)
\(L(\frac12)\) \(\approx\) \(0.7338965933 - 0.8712528109i\)
\(L(1)\) \(\approx\) \(0.7955004026 - 0.3980798159i\)
\(L(1)\) \(\approx\) \(0.7955004026 - 0.3980798159i\)

Euler product

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

−21.34506418454048729671440350829, −20.291108483571658334796326292132, −19.54571371343935546297830471444, −18.71625790312951767007888481192, −18.12027260387162590284078497790, −17.176516815112926585542647721644, −16.83415756938883422521364809742, −15.91138115842584898383955462536, −15.257986977997295268169618881360, −14.19989766773778522338328117886, −13.73617346538223361990692660945, −13.01996777558873632675476364891, −12.078176261680014878984746237637, −10.59347704465206575076440748337, −10.08295387223140189786627544380, −9.538912400789950648691499404026, −8.600468900562180545486632447870, −7.75959550251224421289531685115, −6.73116619468381028790281680475, −6.16756817812279121610037130439, −5.557502419483910419417149293353, −4.41355174428629048354990374876, −3.404790198053390227109349037, −2.09433935205656292383675082483, −1.02625107395456188681589379007, 0.61219297717323239845171311330, 1.80548338182104082128542737740, 2.72117720761391384671808636384, 3.31019212521299241100067656619, 4.62441516196654521044796651974, 5.417067741060299956391640739695, 6.48754170537944145996919689740, 7.32741120006908735595979628256, 8.57595564402911555619512574615, 9.216914567332480833607638224240, 9.84985061362611991929816699433, 10.43411196074885699111038920005, 11.48009579201789405719959786016, 12.32002601938087562571528505836, 12.88467020729695291303124697998, 13.88816409051538570228034132761, 14.08585016556464724231389683944, 15.8594117975032771956272766716, 16.145820654702300698028914865531, 17.35928604615988910354537266557, 17.78531408660489466101029758122, 18.547715966906906852825552683194, 19.29855848864998763766036143112, 20.08536634021338390939630577900, 20.699720423127139136932679853414

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