Properties

Label 1-709-709.11-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.548 - 0.836i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (0.943 − 0.330i)2-s + (0.115 − 0.993i)3-s + (0.781 − 0.624i)4-s + (−0.202 − 0.979i)5-s + (−0.220 − 0.975i)6-s + (0.254 + 0.967i)7-s + (0.530 − 0.847i)8-s + (−0.973 − 0.228i)9-s + (−0.515 − 0.857i)10-s + (0.969 + 0.245i)11-s + (−0.530 − 0.847i)12-s + (0.964 − 0.263i)13-s + (0.560 + 0.828i)14-s + (−0.996 + 0.0886i)15-s + (0.220 − 0.975i)16-s + (0.271 − 0.962i)17-s + ⋯
L(s)  = 1  + (0.943 − 0.330i)2-s + (0.115 − 0.993i)3-s + (0.781 − 0.624i)4-s + (−0.202 − 0.979i)5-s + (−0.220 − 0.975i)6-s + (0.254 + 0.967i)7-s + (0.530 − 0.847i)8-s + (−0.973 − 0.228i)9-s + (−0.515 − 0.857i)10-s + (0.969 + 0.245i)11-s + (−0.530 − 0.847i)12-s + (0.964 − 0.263i)13-s + (0.560 + 0.828i)14-s + (−0.996 + 0.0886i)15-s + (0.220 − 0.975i)16-s + (0.271 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.548 - 0.836i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.548 - 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.356923134 - 2.512843968i\)
\(L(\frac12)\) \(\approx\) \(1.356923134 - 2.512843968i\)
\(L(1)\) \(\approx\) \(1.549218220 - 1.256457533i\)
\(L(1)\) \(\approx\) \(1.549218220 - 1.256457533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.943 - 0.330i)T \)
3 \( 1 + (0.115 - 0.993i)T \)
5 \( 1 + (-0.202 - 0.979i)T \)
7 \( 1 + (0.254 + 0.967i)T \)
11 \( 1 + (0.969 + 0.245i)T \)
13 \( 1 + (0.964 - 0.263i)T \)
17 \( 1 + (0.271 - 0.962i)T \)
19 \( 1 + (0.999 + 0.0354i)T \)
23 \( 1 + (-0.895 - 0.445i)T \)
29 \( 1 + (-0.589 + 0.807i)T \)
31 \( 1 + (-0.421 + 0.906i)T \)
37 \( 1 + (-0.254 - 0.967i)T \)
41 \( 1 + (0.671 - 0.740i)T \)
43 \( 1 + (0.355 + 0.934i)T \)
47 \( 1 + (-0.833 + 0.552i)T \)
53 \( 1 + (-0.574 - 0.818i)T \)
59 \( 1 + (-0.530 + 0.847i)T \)
61 \( 1 + (0.852 - 0.522i)T \)
67 \( 1 + (0.924 + 0.380i)T \)
71 \( 1 + (-0.515 + 0.857i)T \)
73 \( 1 + (-0.388 + 0.921i)T \)
79 \( 1 + (-0.842 + 0.537i)T \)
83 \( 1 + (-0.861 + 0.507i)T \)
89 \( 1 + (0.903 - 0.429i)T \)
97 \( 1 + (0.167 + 0.985i)T \)
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

−22.76548660701539184625968400220, −22.13815328517023131529189843507, −21.519186436502806967108327157452, −20.56029721944553926368282640455, −20.01096339171239309902015593524, −19.0510058217492160383196894492, −17.65862674881963951464284567688, −16.9110820991435132466319711739, −16.17306058137485996876919468436, −15.35150963911243421818236148314, −14.608540194747967052833349117, −13.999340560316193605084553416776, −13.414372984021825962343201863957, −11.75928488968020412203314052568, −11.32080195062749974371376225287, −10.559305200339921404348888219957, −9.62961685974995554877625423370, −8.25329187241353934072183538176, −7.52537888508436691471905097177, −6.37426892829149667747922878456, −5.78972111697407975795983271568, −4.38045853730074284848331764308, −3.75281847313491864093809462039, −3.29564531661534853714241562943, −1.772891625338805277286147668879, 1.0917251193346617929139098455, 1.77294220055023053723111552497, 2.95622117704298891666906092027, 3.96488988483534411069893893679, 5.25167836499317277095388893083, 5.74178791209839978595028277279, 6.78939941815858485382976007159, 7.78509217398399579408857028877, 8.80368999666288425323465593462, 9.53533021242496720423104387085, 11.22530263108268010438299774417, 11.747613552551275331790832114058, 12.48040415222472539474429279135, 12.95906809712557343138404347119, 14.136821861409589333006010845383, 14.45686488940579882601320137594, 15.88255461888278429554950648299, 16.2244706700605614686176202072, 17.64606967763595017177072272353, 18.37086106514219177701767346948, 19.27202100802336461787887512520, 20.13447657298235200725946260250, 20.52613538532335835545455050192, 21.50068206676196437149405545515, 22.55961242906854329134642106474

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