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Properties

Label 1-709-709.129-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

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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: $\chi_{709} (129, \cdot )$
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 \( 1 + (0.167 - 0.985i)T \)
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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

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

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