↑

Properties

Label 1-283-283.267-r1-0-0
Degree $1$
Conductor $283$
Sign $0.958 - 0.285i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.824 + 0.565i)2-s + (0.0334 − 0.999i)3-s + (0.359 + 0.933i)4-s + (0.997 − 0.0667i)5-s + (0.593 − 0.805i)6-s + (0.991 − 0.133i)7-s + (−0.231 + 0.972i)8-s + (−0.997 − 0.0667i)9-s + (0.860 + 0.509i)10-s + (0.359 − 0.933i)11-s + (0.944 − 0.328i)12-s + (0.593 − 0.805i)13-s + (0.892 + 0.451i)14-s + (−0.0334 − 0.999i)15-s + (−0.741 + 0.670i)16-s + (0.892 − 0.451i)17-s + ⋯
L(s)  = 1  + (0.824 + 0.565i)2-s + (0.0334 − 0.999i)3-s + (0.359 + 0.933i)4-s + (0.997 − 0.0667i)5-s + (0.593 − 0.805i)6-s + (0.991 − 0.133i)7-s + (−0.231 + 0.972i)8-s + (−0.997 − 0.0667i)9-s + (0.860 + 0.509i)10-s + (0.359 − 0.933i)11-s + (0.944 − 0.328i)12-s + (0.593 − 0.805i)13-s + (0.892 + 0.451i)14-s + (−0.0334 − 0.999i)15-s + (−0.741 + 0.670i)16-s + (0.892 − 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.958 - 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.339535746 - 0.6335419589i\)
\(L(\frac12)\) \(\approx\) \(4.339535746 - 0.6335419589i\)
\(L(1)\) \(\approx\) \(2.262928030 - 0.06540955458i\)
\(L(1)\) \(\approx\) \(2.262928030 - 0.06540955458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.824 + 0.565i)T \)
3 \( 1 + (0.0334 - 0.999i)T \)
5 \( 1 + (0.997 - 0.0667i)T \)
7 \( 1 + (0.991 - 0.133i)T \)
11 \( 1 + (0.359 - 0.933i)T \)
13 \( 1 + (0.593 - 0.805i)T \)
17 \( 1 + (0.892 - 0.451i)T \)
19 \( 1 + (-0.593 + 0.805i)T \)
23 \( 1 + (-0.979 - 0.199i)T \)
29 \( 1 + (0.784 - 0.619i)T \)
31 \( 1 + (0.420 + 0.907i)T \)
37 \( 1 + (-0.920 - 0.390i)T \)
41 \( 1 + (0.231 + 0.972i)T \)
43 \( 1 + (-0.695 - 0.718i)T \)
47 \( 1 + (-0.593 - 0.805i)T \)
53 \( 1 + (0.741 + 0.670i)T \)
59 \( 1 + (0.964 - 0.264i)T \)
61 \( 1 + (0.860 - 0.509i)T \)
67 \( 1 + (0.944 + 0.328i)T \)
71 \( 1 + (0.359 - 0.933i)T \)
73 \( 1 + (-0.420 + 0.907i)T \)
79 \( 1 + (-0.695 + 0.718i)T \)
83 \( 1 + (0.100 - 0.994i)T \)
89 \( 1 + (0.695 + 0.718i)T \)
97 \( 1 + (-0.979 - 0.199i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.512160295605779801099337865489, −24.31947604410251091642658129916, −23.41974391459500647360000277974, −22.38560820010149317145318123869, −21.607094558958407400565338178608, −21.05411254393772976895232920820, −20.45237791945313841256544041382, −19.30441883938698808003891907480, −17.99752711711994966872545159885, −17.16029099715275219690578569355, −15.96272596695993013161313342725, −14.84096579490212997135016715432, −14.38064599620171022378952511435, −13.5123182660934021620942101997, −12.16087197284334841345071061405, −11.29138261796003361520783369801, −10.3379922751091996858027497788, −9.63830053249226598617984443798, −8.5583954788868086515229507648, −6.681025544992582957372507303724, −5.6253675363169906169617798179, −4.754557600270069740849366206396, −3.91322424888243179724747933281, −2.444702448743896043779793352492, −1.508384473609972114884340928031, 1.13403169037126904505130904791, 2.31624840944886453278751609732, 3.55603467521125848764653563955, 5.231353927535254645869452221049, 5.88540996060012486050817136437, 6.765068334046041622198947995091, 8.134571266265372147562813973662, 8.48653014740237062747729273043, 10.41205032516756734349536924426, 11.59856312080677631853639537880, 12.41669647049243744802901197895, 13.5029439015809251238055383621, 14.046228316115763505312956900696, 14.669984940770244774041264204029, 16.20726942130885862129370798239, 17.142204068496527777977400160153, 17.81240149962965853260541801205, 18.64114130581433541252999032398, 20.12806768064576882434283756316, 20.990349115042507236048248926025, 21.681632068746319179734902162178, 22.90293059837209388331511760944, 23.54115584998540774539240618679, 24.70177434122689184023558574494, 24.86142410560188738059639194632

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