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Properties

Label 1-709-709.473-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.621 + 0.783i$
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  + (0.0443 − 0.999i)2-s + (0.220 + 0.975i)3-s + (−0.996 − 0.0886i)4-s + (0.150 − 0.988i)5-s + (0.984 − 0.176i)6-s + (−0.671 + 0.740i)7-s + (−0.132 + 0.991i)8-s + (−0.903 + 0.429i)9-s + (−0.981 − 0.194i)10-s + (0.115 + 0.993i)11-s + (−0.132 − 0.991i)12-s + (0.977 − 0.211i)13-s + (0.710 + 0.703i)14-s + (0.997 − 0.0709i)15-s + (0.984 + 0.176i)16-s + (−0.917 + 0.396i)17-s + ⋯
L(s)  = 1  + (0.0443 − 0.999i)2-s + (0.220 + 0.975i)3-s + (−0.996 − 0.0886i)4-s + (0.150 − 0.988i)5-s + (0.984 − 0.176i)6-s + (−0.671 + 0.740i)7-s + (−0.132 + 0.991i)8-s + (−0.903 + 0.429i)9-s + (−0.981 − 0.194i)10-s + (0.115 + 0.993i)11-s + (−0.132 − 0.991i)12-s + (0.977 − 0.211i)13-s + (0.710 + 0.703i)14-s + (0.997 − 0.0709i)15-s + (0.984 + 0.176i)16-s + (−0.917 + 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.621 + 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1179182082 + 0.2439758472i\)
\(L(\frac12)\) \(\approx\) \(0.1179182082 + 0.2439758472i\)
\(L(1)\) \(\approx\) \(0.7395099475 - 0.1020129954i\)
\(L(1)\) \(\approx\) \(0.7395099475 - 0.1020129954i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.0443 - 0.999i)T \)
3 \( 1 + (0.220 + 0.975i)T \)
5 \( 1 + (0.150 - 0.988i)T \)
7 \( 1 + (-0.671 + 0.740i)T \)
11 \( 1 + (0.115 + 0.993i)T \)
13 \( 1 + (0.977 - 0.211i)T \)
17 \( 1 + (-0.917 + 0.396i)T \)
19 \( 1 + (-0.792 - 0.610i)T \)
23 \( 1 + (0.631 - 0.775i)T \)
29 \( 1 + (-0.992 + 0.123i)T \)
31 \( 1 + (-0.560 - 0.828i)T \)
37 \( 1 + (-0.671 + 0.740i)T \)
41 \( 1 + (-0.271 + 0.962i)T \)
43 \( 1 + (-0.943 - 0.330i)T \)
47 \( 1 + (-0.987 - 0.159i)T \)
53 \( 1 + (-0.437 + 0.899i)T \)
59 \( 1 + (-0.132 + 0.991i)T \)
61 \( 1 + (0.684 + 0.728i)T \)
67 \( 1 + (-0.589 - 0.807i)T \)
71 \( 1 + (-0.981 + 0.194i)T \)
73 \( 1 + (0.949 + 0.314i)T \)
79 \( 1 + (-0.468 + 0.883i)T \)
83 \( 1 + (0.910 - 0.413i)T \)
89 \( 1 + (0.937 - 0.347i)T \)
97 \( 1 + (-0.722 + 0.691i)T \)
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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

−22.65974137499431748605722489764, −21.8416494178807921555716473960, −20.70504241517052626652833447916, −19.26047353350155717035735932134, −19.12726797702261057987944005071, −18.16579615180074202177208812721, −17.50784817331680825481843315768, −16.60561861754718962073892821247, −15.794515227393670613737320669904, −14.74292664056557223474467587853, −14.00244636949046190112483986807, −13.43714891233611012350294102943, −12.86930447746196776045346387952, −11.40173344807871714769805071351, −10.64877490686399800108133198232, −9.37109088987134442745394553589, −8.58440985712325932999146416918, −7.6264973527506109924056067909, −6.74758795882264979775520318893, −6.433483538921395275822236344, −5.49337862661349210413340094333, −3.70984771910722633622746600399, −3.31279770972049108376440186498, −1.67244504583735012253932458977, −0.11975993605598443714706184508, 1.7153828475340024253747924218, 2.642767455204788709201728221665, 3.77539211961377859249045007009, 4.53405633717111450544358327474, 5.29167796980503211782842438825, 6.33491203884985370977542544066, 8.27231172125302959748025676935, 8.90888961852043291830680668114, 9.39629839397589208578500427725, 10.28846120242456697854165819341, 11.15909343716323607903427929, 12.06575400157207821869164464944, 13.09445335061662639668008165303, 13.31513314666887645572114316615, 14.91367827036873344445419304496, 15.289802104264157547763630091729, 16.44640913274168734080355984773, 17.15986663764086369885857266953, 18.09380780130776972562806394975, 19.10365662109709287045191999735, 20.02731956635516733537470422512, 20.4162324333457246452304163826, 21.195373553954497846991046593899, 21.91541629424051180496157059929, 22.59907518405828616947595020967

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