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Properties

Label 1-783-783.211-r1-0-0
Degree $1$
Conductor $783$
Sign $-0.977 - 0.212i$
Analytic cond. $84.1450$
Root an. cond. $84.1450$
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  + (0.749 − 0.661i)2-s + (0.124 − 0.992i)4-s + (0.0249 + 0.999i)5-s + (−0.124 − 0.992i)7-s + (−0.563 − 0.826i)8-s + (0.680 + 0.733i)10-s + (0.246 − 0.969i)11-s + (0.583 + 0.811i)13-s + (−0.749 − 0.661i)14-s + (−0.969 − 0.246i)16-s + (0.866 − 0.5i)17-s + (−0.294 + 0.955i)19-s + (0.995 + 0.0995i)20-s + (−0.456 − 0.889i)22-s + (−0.318 − 0.947i)23-s + ⋯
L(s)  = 1  + (0.749 − 0.661i)2-s + (0.124 − 0.992i)4-s + (0.0249 + 0.999i)5-s + (−0.124 − 0.992i)7-s + (−0.563 − 0.826i)8-s + (0.680 + 0.733i)10-s + (0.246 − 0.969i)11-s + (0.583 + 0.811i)13-s + (−0.749 − 0.661i)14-s + (−0.969 − 0.246i)16-s + (0.866 − 0.5i)17-s + (−0.294 + 0.955i)19-s + (0.995 + 0.0995i)20-s + (−0.456 − 0.889i)22-s + (−0.318 − 0.947i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(84.1450\)
Root analytic conductor: \(84.1450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 783,\ (1:\ ),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2452021605 - 2.278841178i\)
\(L(\frac12)\) \(\approx\) \(0.2452021605 - 2.278841178i\)
\(L(1)\) \(\approx\) \(1.241898641 - 0.7974038810i\)
\(L(1)\) \(\approx\) \(1.241898641 - 0.7974038810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.749 - 0.661i)T \)
5 \( 1 + (0.0249 + 0.999i)T \)
7 \( 1 + (-0.124 - 0.992i)T \)
11 \( 1 + (0.246 - 0.969i)T \)
13 \( 1 + (0.583 + 0.811i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.294 + 0.955i)T \)
23 \( 1 + (-0.318 - 0.947i)T \)
31 \( 1 + (-0.521 - 0.853i)T \)
37 \( 1 + (0.997 - 0.0747i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.999 - 0.0249i)T \)
47 \( 1 + (-0.715 - 0.698i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.992 - 0.124i)T \)
67 \( 1 + (-0.270 + 0.962i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (0.149 + 0.988i)T \)
79 \( 1 + (-0.811 - 0.583i)T \)
83 \( 1 + (-0.998 + 0.0498i)T \)
89 \( 1 + (-0.930 + 0.365i)T \)
97 \( 1 + (-0.889 - 0.456i)T \)
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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

−22.60942612415328409910473870556, −21.62771321108996839982382689578, −21.20227306665295711599853209880, −20.217759291053535329219784634692, −19.54830402377978297895967454197, −18.0739827154244098018602383208, −17.61921226355899118036135187089, −16.66089161304178477122393365398, −15.86595935881276123608346373423, −15.25933038571651659585755372965, −14.549035861154606238271079858551, −13.30816218626355234251349580557, −12.78398564055787404954913226579, −12.158265572429176016824151428513, −11.32686895324685473429594882570, −9.82494197274256100364104505664, −8.94573487003310859221268452727, −8.22765810210248288328682191610, −7.36735630859434864278949862901, −6.14108306620270480003558693638, −5.49487873501355187845572819318, −4.73329256106230811305144403218, −3.7308356797938157250913821859, −2.63202634491654556791375343609, −1.40458308675194830019232121799, 0.38010306952339767301688499156, 1.53833467435573832697619131627, 2.720373313077583811517268690648, 3.6959157472476337775361904412, 4.1300412654261796733378967079, 5.63425003941228713750956687497, 6.36776690108091113375186148394, 7.126819404002593084882900000582, 8.33436418216478388637504071838, 9.73080587410450146610672110051, 10.26189076291320087275069069923, 11.226470805331282849458076164189, 11.578702628326706288033711782346, 12.85992745129591281002257304463, 13.687206187680204558055229120889, 14.32950031103504515146166811000, 14.74675136899618705442864912326, 16.20075244061608856731859159120, 16.59078258921962739492863246834, 18.084646667610556708233380997206, 18.80811782288292471402302523405, 19.26708717238953446088736835249, 20.31653736290519261654366279770, 21.02224742877021851798321391334, 21.75717763829428368799007031628

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