↑

Properties

Label 1-709-709.29-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.197 + 0.980i$
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  + (0.823 + 0.567i)2-s + (−0.992 − 0.123i)3-s + (0.355 + 0.934i)4-s + (0.895 − 0.445i)5-s + (−0.746 − 0.665i)6-s + (0.999 − 0.0354i)7-s + (−0.237 + 0.971i)8-s + (0.969 + 0.245i)9-s + (0.989 + 0.141i)10-s + (−0.589 + 0.807i)11-s + (−0.237 − 0.971i)12-s + (−0.530 + 0.847i)13-s + (0.842 + 0.537i)14-s + (−0.943 + 0.330i)15-s + (−0.746 + 0.665i)16-s + (−0.405 + 0.914i)17-s + ⋯
L(s)  = 1  + (0.823 + 0.567i)2-s + (−0.992 − 0.123i)3-s + (0.355 + 0.934i)4-s + (0.895 − 0.445i)5-s + (−0.746 − 0.665i)6-s + (0.999 − 0.0354i)7-s + (−0.237 + 0.971i)8-s + (0.969 + 0.245i)9-s + (0.989 + 0.141i)10-s + (−0.589 + 0.807i)11-s + (−0.237 − 0.971i)12-s + (−0.530 + 0.847i)13-s + (0.842 + 0.537i)14-s + (−0.943 + 0.330i)15-s + (−0.746 + 0.665i)16-s + (−0.405 + 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244073556 + 1.519143224i\)
\(L(\frac12)\) \(\approx\) \(1.244073556 + 1.519143224i\)
\(L(1)\) \(\approx\) \(1.308396233 + 0.6806273723i\)
\(L(1)\) \(\approx\) \(1.308396233 + 0.6806273723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.823 + 0.567i)T \)
3 \( 1 + (-0.992 - 0.123i)T \)
5 \( 1 + (0.895 - 0.445i)T \)
7 \( 1 + (0.999 - 0.0354i)T \)
11 \( 1 + (-0.589 + 0.807i)T \)
13 \( 1 + (-0.530 + 0.847i)T \)
17 \( 1 + (-0.405 + 0.914i)T \)
19 \( 1 + (-0.722 - 0.691i)T \)
23 \( 1 + (0.879 + 0.476i)T \)
29 \( 1 + (-0.981 - 0.194i)T \)
31 \( 1 + (0.758 + 0.651i)T \)
37 \( 1 + (0.999 - 0.0354i)T \)
41 \( 1 + (-0.792 + 0.610i)T \)
43 \( 1 + (0.684 + 0.728i)T \)
47 \( 1 + (-0.0266 - 0.999i)T \)
53 \( 1 + (0.185 - 0.982i)T \)
59 \( 1 + (-0.237 + 0.971i)T \)
61 \( 1 + (-0.671 + 0.740i)T \)
67 \( 1 + (0.515 - 0.857i)T \)
71 \( 1 + (0.989 - 0.141i)T \)
73 \( 1 + (-0.998 + 0.0532i)T \)
79 \( 1 + (0.937 + 0.347i)T \)
83 \( 1 + (-0.437 - 0.899i)T \)
89 \( 1 + (0.115 + 0.993i)T \)
97 \( 1 + (-0.954 - 0.297i)T \)
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

−22.29561214663600025990494063264, −21.68476027836689832952545052235, −20.9154448975687745207604699936, −20.47447279348114369162366252694, −18.75230414419991351859189218337, −18.55933925551679365069295486543, −17.490025907500960743746745264358, −16.79695169993991279675223225112, −15.58250335190529347192116852756, −14.89602669306522597937571629814, −14.01704888106729468632378394654, −13.17405656825212298132603942410, −12.45267618833758008886536102599, −11.35043663659193492488418847065, −10.856594283855329907722902766204, −10.24917512881762807101195682730, −9.23997189325856417682530801256, −7.68789105014214926034933125249, −6.618388537537388559262558894652, −5.68625672841793179563012130147, −5.235479285287127328572483058053, −4.33012287763941253944174965623, −2.93589957178962933820788820618, −2.03803315128849606869288659278, −0.80758145085733786279384050466, 1.61506540522660137014424657496, 2.34997189979060154720011528155, 4.32036182013298520343212869080, 4.78660331148426308090125172550, 5.49317453570493015595978894324, 6.4734528367966306543008521271, 7.19080442500707429453578589112, 8.22320273870672331315602670191, 9.34204828238149419661640083875, 10.54294011546812464289337666163, 11.3358361341112496760088227915, 12.20660228655713701836968690552, 13.01808829538813713044265538282, 13.53183730882309586997032693579, 14.77862340456146649632497150224, 15.26545439971867018128553344695, 16.50501421172790544444425210017, 17.12869692820826295935218253143, 17.5823012777258448221972959005, 18.3184915531728899691942486758, 19.76783896412053715917131132209, 20.998617864054891344236191176836, 21.36723283076846762728996406854, 21.92939793550733415752410254890, 22.99871914226158812727769887015

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