↑

Properties

Label 1-1063-1063.2-r0-0-0
Degree $1$
Conductor $1063$
Sign $0.647 + 0.762i$
Analytic cond. $4.93655$
Root an. cond. $4.93655$
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  + (0.399 + 0.916i)2-s + (0.996 − 0.0827i)3-s + (−0.680 + 0.732i)4-s + (−0.722 − 0.691i)5-s + (0.474 + 0.880i)6-s + (−0.939 + 0.342i)7-s + (−0.943 − 0.330i)8-s + (0.986 − 0.164i)9-s + (0.344 − 0.938i)10-s + (−0.976 − 0.217i)11-s + (−0.617 + 0.786i)12-s + (0.631 − 0.775i)13-s + (−0.689 − 0.724i)14-s + (−0.777 − 0.629i)15-s + (−0.0738 − 0.997i)16-s + (0.658 + 0.752i)17-s + ⋯
L(s)  = 1  + (0.399 + 0.916i)2-s + (0.996 − 0.0827i)3-s + (−0.680 + 0.732i)4-s + (−0.722 − 0.691i)5-s + (0.474 + 0.880i)6-s + (−0.939 + 0.342i)7-s + (−0.943 − 0.330i)8-s + (0.986 − 0.164i)9-s + (0.344 − 0.938i)10-s + (−0.976 − 0.217i)11-s + (−0.617 + 0.786i)12-s + (0.631 − 0.775i)13-s + (−0.689 − 0.724i)14-s + (−0.777 − 0.629i)15-s + (−0.0738 − 0.997i)16-s + (0.658 + 0.752i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ 0.647 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.710077412 + 0.7911372720i\)
\(L(\frac12)\) \(\approx\) \(1.710077412 + 0.7911372720i\)
\(L(1)\) \(\approx\) \(1.280881798 + 0.4980869016i\)
\(L(1)\) \(\approx\) \(1.280881798 + 0.4980869016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.399 + 0.916i)T \)
3 \( 1 + (0.996 - 0.0827i)T \)
5 \( 1 + (-0.722 - 0.691i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.976 - 0.217i)T \)
13 \( 1 + (0.631 - 0.775i)T \)
17 \( 1 + (0.658 + 0.752i)T \)
19 \( 1 + (0.999 + 0.0118i)T \)
23 \( 1 + (0.196 + 0.980i)T \)
29 \( 1 + (-0.540 - 0.841i)T \)
31 \( 1 + (-0.569 - 0.821i)T \)
37 \( 1 + (0.603 + 0.797i)T \)
41 \( 1 + (0.937 - 0.347i)T \)
43 \( 1 + (0.758 + 0.651i)T \)
47 \( 1 + (0.929 - 0.369i)T \)
53 \( 1 + (0.999 - 0.0236i)T \)
59 \( 1 + (0.545 - 0.838i)T \)
61 \( 1 + (0.836 - 0.547i)T \)
67 \( 1 + (0.0915 + 0.995i)T \)
71 \( 1 + (-0.887 - 0.461i)T \)
73 \( 1 + (-0.394 + 0.918i)T \)
79 \( 1 + (-0.994 + 0.100i)T \)
83 \( 1 + (0.726 - 0.686i)T \)
89 \( 1 + (-0.0974 + 0.995i)T \)
97 \( 1 + (0.505 - 0.863i)T \)
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

−21.19621490259121561317600094204, −20.52071894561034281134058652686, −19.923480089210503019272121402883, −19.19590168916292588854662016446, −18.50054175286331580899559902183, −18.201761315336863097875940601005, −16.14726724166464621511526450486, −15.99465551270058960094890222238, −14.80233606633469807567124766821, −14.23598286542345042057603967084, −13.514714257297753203757188371717, −12.74111305683137084635596487735, −12.00092544147121300253904391417, −10.843362822319281273773588813341, −10.36866273028517567340796164941, −9.44441765080263895732396111889, −8.78529679773181438451081400926, −7.54003021062817828915275289783, −6.97957445706391720497626170992, −5.66718872526965345360911177104, −4.439225880738142674057483912659, −3.703311954853222306761273650009, −3.002303137032554607745277849599, −2.39886052820443882725900738577, −0.91734931210859120118339901563, 0.84950363765266714130803308299, 2.67569180268505549235207875217, 3.50069483463335839277170205106, 4.02815174565990502762649145239, 5.37947529784215150936603472017, 5.91609773465355382731456560990, 7.34951117467096804015936773149, 7.78726299443947171482836693266, 8.486660519075390262310264618076, 9.30737545326107489435548134781, 10.035501992460302398296751509791, 11.55203748057446558830632332698, 12.64181269285801069268666264585, 13.06048524595989671513406128398, 13.568252767896069099468973819467, 14.772412646914601692879887846928, 15.4898307500378709130104277891, 15.85505634618290029204626808881, 16.5316954869048618040201563783, 17.66317795938109846662430622310, 18.70155881021354379188763229645, 19.08580647339789603549520258224, 20.19891716877860259598982071260, 20.78842484998365673100400924342, 21.59369019324068552651350713248

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