Properties

Degree $1$
Conductor $283$
Sign $0.958 + 0.285i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.958 + 0.285i$
Motivic weight: \(0\)
Character: $\chi_{283} (53, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.958 + 0.285i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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