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Properties

Label 1-1287-1287.1163-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.831 - 0.555i$
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  + (0.743 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9607830447 - 3.165131992i\)
\(L(\frac12)\) \(\approx\) \(0.9607830447 - 3.165131992i\)
\(L(1)\) \(\approx\) \(1.303436192 - 0.9883013513i\)
\(L(1)\) \(\approx\) \(1.303436192 - 0.9883013513i\)

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

−21.12072196163265208061489282984, −20.72913514147178505975193953118, −19.51735194741150749204493310153, −18.89213249651051096739313911777, −17.75276668640009044087869735462, −17.36201478791140975793511552894, −16.27035073796950375252163737991, −15.85390567549354695670659451712, −14.854110623136744579162484197685, −14.04269148402019723099037332691, −13.815218773356434575517488956460, −12.75328115218672320648549164299, −12.00634262374384411699007375888, −11.03930571268653979739850211602, −10.25646546303130951692157214271, −9.40406923697097885696870154522, −8.236676308906188257054988753172, −7.13398075791364893493347278373, −7.05658741623343903334771803229, −5.96928823024977352639647715798, −5.19802666022111852257402192361, −4.07236800707092282838981129139, −3.2253622382586021996227043715, −2.72937302037187120390779262782, −1.06233923140704293018332704927, 0.55796604963211883318070745341, 1.38198529071448683536029014741, 2.52628579938571604283813985328, 3.27670978697529581739283813517, 4.33888718426883482486961031543, 5.24835837453130509201964242224, 5.74209216009934192633439602199, 6.65750446007801851816691508897, 7.94088105397735872583197254077, 9.01565416696037191462219953721, 9.56223732887614488389587045354, 10.30030375273024880224429516450, 11.46321937703039025266676495931, 12.18629236488115258350494700148, 12.60333557033686176021558139030, 13.46770774495543011263043922424, 14.09132285904093491486531883862, 15.174485623981622467193463685080, 15.76397932144397096110579234524, 16.51711867427811795071110150077, 17.495292938628573493441304490492, 18.49612264617229013999412237998, 19.19507738156546407168602545329, 19.79669328448609838976210315153, 20.74808411479777672548298243322

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