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Properties

Label 1-283-283.2-r1-0-0
Degree $1$
Conductor $283$
Sign $0.836 + 0.547i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (−0.593 − 0.805i)2-s + (−0.920 − 0.390i)3-s + (−0.296 + 0.955i)4-s + (−0.695 + 0.718i)5-s + (0.231 + 0.972i)6-s + (−0.0334 − 0.999i)7-s + (0.944 − 0.328i)8-s + (0.695 + 0.718i)9-s + (0.991 + 0.133i)10-s + (−0.296 − 0.955i)11-s + (0.645 − 0.763i)12-s + (0.231 + 0.972i)13-s + (−0.784 + 0.619i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (−0.784 − 0.619i)17-s + ⋯
L(s)  = 1  + (−0.593 − 0.805i)2-s + (−0.920 − 0.390i)3-s + (−0.296 + 0.955i)4-s + (−0.695 + 0.718i)5-s + (0.231 + 0.972i)6-s + (−0.0334 − 0.999i)7-s + (0.944 − 0.328i)8-s + (0.695 + 0.718i)9-s + (0.991 + 0.133i)10-s + (−0.296 − 0.955i)11-s + (0.645 − 0.763i)12-s + (0.231 + 0.972i)13-s + (−0.784 + 0.619i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (−0.784 − 0.619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2189603924 + 0.06526674253i\)
\(L(\frac12)\) \(\approx\) \(0.2189603924 + 0.06526674253i\)
\(L(1)\) \(\approx\) \(0.3893900587 - 0.1889998116i\)
\(L(1)\) \(\approx\) \(0.3893900587 - 0.1889998116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.593 - 0.805i)T \)
3 \( 1 + (-0.920 - 0.390i)T \)
5 \( 1 + (-0.695 + 0.718i)T \)
7 \( 1 + (-0.0334 - 0.999i)T \)
11 \( 1 + (-0.296 - 0.955i)T \)
13 \( 1 + (0.231 + 0.972i)T \)
17 \( 1 + (-0.784 - 0.619i)T \)
19 \( 1 + (-0.231 - 0.972i)T \)
23 \( 1 + (-0.741 + 0.670i)T \)
29 \( 1 + (-0.166 - 0.986i)T \)
31 \( 1 + (-0.480 - 0.876i)T \)
37 \( 1 + (-0.100 + 0.994i)T \)
41 \( 1 + (-0.944 - 0.328i)T \)
43 \( 1 + (0.979 + 0.199i)T \)
47 \( 1 + (-0.231 + 0.972i)T \)
53 \( 1 + (0.824 - 0.565i)T \)
59 \( 1 + (-0.997 + 0.0667i)T \)
61 \( 1 + (0.991 - 0.133i)T \)
67 \( 1 + (0.645 + 0.763i)T \)
71 \( 1 + (-0.296 - 0.955i)T \)
73 \( 1 + (0.480 - 0.876i)T \)
79 \( 1 + (0.979 - 0.199i)T \)
83 \( 1 + (0.359 + 0.933i)T \)
89 \( 1 + (-0.979 - 0.199i)T \)
97 \( 1 + (-0.741 + 0.670i)T \)
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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

−25.142059216232301931450478086384, −24.434699890579526252519140752558, −23.46904598594310496530290231113, −22.867317597730680803036274402659, −21.91835267801844869568553735909, −20.58871523883249353837919397618, −19.762849264011805234116381714244, −18.43555680429239208663374246881, −17.92379587150536370377245941171, −16.91638859768186616615564404741, −16.013447138746409161502363508596, −15.460153900304690135720424377986, −14.73808041375679603279408285907, −12.78921930039494707154266316296, −12.291617558961890727085492881721, −10.93764018636866356275407346465, −10.09446379319469585856926476191, −8.95061608499577187691066406289, −8.157287057835869843232269208800, −6.94526700465288230964193023533, −5.75656515887241215430163923007, −5.10368265240326400876779457824, −4.01628526700549473131109659862, −1.70458310917434378794804008669, −0.15341873598968580007453650323, 0.72484908733137925235845147757, 2.26735783013002773480286135884, 3.7078691689215316953711479322, 4.58544040857127066820621985612, 6.43763318303153017676050560774, 7.253750580997286427098630848567, 8.1173347449521516329022937270, 9.584257488489985233550019976976, 10.753932136373422452673471618302, 11.25215597406056128352353006933, 11.84334657794374986314181857754, 13.333030178679834610712992701238, 13.778237801811293291544189074601, 15.70832931794450168546328072907, 16.47778024124490318691892057182, 17.37227776550688230831440979168, 18.25803609787001600902475823990, 19.058906245827872761598824590167, 19.661020014909941262515904828182, 20.86752973792134171048139697533, 21.982024610569102846416179306940, 22.51872982580268858264721135535, 23.63889591464298413559882866113, 24.144501201286426109946034716668, 25.899977782853536494492541424341

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