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Properties

Label 1-2563-2563.2562-r1-0-0
Degree $1$
Conductor $2563$
Sign $1$
Analytic cond. $275.432$
Root an. cond. $275.432$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 29-s − 30-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 29-s − 30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2563\)    =    \(11 \cdot 233\)
Sign: $1$
Analytic conductor: \(275.432\)
Root analytic conductor: \(275.432\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2563} (2562, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2563,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4042508979\)
\(L(\frac12)\) \(\approx\) \(0.4042508979\)
\(L(1)\) \(\approx\) \(0.3723289623\)
\(L(1)\) \(\approx\) \(0.3723289623\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
233 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−19.19482468416610875617477114616, −18.70616580160130273570488352956, −17.72048140587037895059170370836, −16.93929880777659425147306721361, −16.586150151297034677414787952265, −15.96598266782530668588918669564, −15.19108482285975675672426015066, −14.69105093330648209232915826021, −13.06248460053664263291303959848, −12.41109869292139377044681881875, −12.06076485723875748276801534394, −11.057379829736102085775107495, −10.703531812039621102443745975050, −9.68284966208587431798715493183, −9.33986455567088241331966828090, −8.0588477822265724743805600088, −7.518541035500213191049812501494, −6.775374286050887942839855006784, −6.19430982767539605989541139627, −5.22739327607547101435743033605, −4.238185214142470412513007835815, −3.29706102653686504484419947968, −2.42283051010483338701862714385, −1.06605983484682401728166867484, −0.369099806631099265694989969262, 0.369099806631099265694989969262, 1.06605983484682401728166867484, 2.42283051010483338701862714385, 3.29706102653686504484419947968, 4.238185214142470412513007835815, 5.22739327607547101435743033605, 6.19430982767539605989541139627, 6.775374286050887942839855006784, 7.518541035500213191049812501494, 8.0588477822265724743805600088, 9.33986455567088241331966828090, 9.68284966208587431798715493183, 10.703531812039621102443745975050, 11.057379829736102085775107495, 12.06076485723875748276801534394, 12.41109869292139377044681881875, 13.06248460053664263291303959848, 14.69105093330648209232915826021, 15.19108482285975675672426015066, 15.96598266782530668588918669564, 16.586150151297034677414787952265, 16.93929880777659425147306721361, 17.72048140587037895059170370836, 18.70616580160130273570488352956, 19.19482468416610875617477114616

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