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Properties

Label 1-709-709.6-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.508 - 0.860i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (0.0354 + 0.999i)2-s + (0.984 + 0.176i)3-s + (−0.997 + 0.0709i)4-s + (−0.421 − 0.906i)5-s + (−0.141 + 0.989i)6-s + (−0.271 − 0.962i)7-s + (−0.106 − 0.994i)8-s + (0.937 + 0.347i)9-s + (0.891 − 0.453i)10-s + (−0.220 − 0.975i)11-s + (−0.994 − 0.106i)12-s + (0.716 − 0.697i)13-s + (0.952 − 0.305i)14-s + (−0.254 − 0.967i)15-s + (0.989 − 0.141i)16-s + (−0.297 − 0.954i)17-s + ⋯
L(s)  = 1  + (0.0354 + 0.999i)2-s + (0.984 + 0.176i)3-s + (−0.997 + 0.0709i)4-s + (−0.421 − 0.906i)5-s + (−0.141 + 0.989i)6-s + (−0.271 − 0.962i)7-s + (−0.106 − 0.994i)8-s + (0.937 + 0.347i)9-s + (0.891 − 0.453i)10-s + (−0.220 − 0.975i)11-s + (−0.994 − 0.106i)12-s + (0.716 − 0.697i)13-s + (0.952 − 0.305i)14-s + (−0.254 − 0.967i)15-s + (0.989 − 0.141i)16-s + (−0.297 − 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.508 - 0.860i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.508 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5778655536 - 1.012806866i\)
\(L(\frac12)\) \(\approx\) \(0.5778655536 - 1.012806866i\)
\(L(1)\) \(\approx\) \(1.087264243 + 0.05680081011i\)
\(L(1)\) \(\approx\) \(1.087264243 + 0.05680081011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.0354 + 0.999i)T \)
3 \( 1 + (0.984 + 0.176i)T \)
5 \( 1 + (-0.421 - 0.906i)T \)
7 \( 1 + (-0.271 - 0.962i)T \)
11 \( 1 + (-0.220 - 0.975i)T \)
13 \( 1 + (0.716 - 0.697i)T \)
17 \( 1 + (-0.297 - 0.954i)T \)
19 \( 1 + (-0.405 + 0.914i)T \)
23 \( 1 + (0.651 - 0.758i)T \)
29 \( 1 + (-0.746 - 0.665i)T \)
31 \( 1 + (-0.703 - 0.710i)T \)
37 \( 1 + (-0.962 + 0.271i)T \)
41 \( 1 + (-0.396 + 0.917i)T \)
43 \( 1 + (-0.0443 + 0.999i)T \)
47 \( 1 + (-0.185 + 0.982i)T \)
53 \( 1 + (0.263 - 0.964i)T \)
59 \( 1 + (0.994 - 0.106i)T \)
61 \( 1 + (-0.567 + 0.823i)T \)
67 \( 1 + (-0.992 + 0.123i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.364 - 0.931i)T \)
79 \( 1 + (0.380 + 0.924i)T \)
83 \( 1 + (-0.999 - 0.0266i)T \)
89 \( 1 + (-0.280 + 0.959i)T \)
97 \( 1 + (-0.0177 - 0.999i)T \)
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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

−22.42935549449829796078604357714, −21.65621865302305972366529691663, −21.14761799511333361551070441236, −20.028636822593723106823287219269, −19.50110128380371296926943347538, −18.7253062053910668509576407952, −18.31994902001009773766242066488, −17.3651567503163137369350928420, −15.63339870511324332723876339784, −15.15840481717726933764041671005, −14.41122380858151498727032163387, −13.45114378993542508392130730288, −12.708618160578588971709286994568, −11.94263062224791303944568322582, −10.93064872918913780546493629193, −10.18721710377413937197014046522, −9.00583198256480581630550366191, −8.774932907248749615309133579826, −7.45343977988003632898589060241, −6.61037318544282953629760410041, −5.18055834644904669540512953971, −3.91722625559126192422276321038, −3.35572188238952899773936521796, −2.268480260838609788970507276211, −1.74960666857456059414759412170, 0.239784907981582544046417343404, 1.20510888305131117744797992331, 3.1673135632862391545998358675, 3.90109214208261910146236780831, 4.69591946346654157912584283473, 5.776047298266649770960180799363, 6.94294864008945915128920911101, 7.920548760057161182888546115084, 8.317824394179805892094922113774, 9.20200659563062720363201632057, 10.035484515588584983544397059457, 11.136712427014677890268359938876, 12.7527920410921763386254568518, 13.24215129749542521885155543885, 13.84919670940126148337503039377, 14.80824622519556773872228476457, 15.64914118311478942133520039651, 16.4415765876366520290079807880, 16.70527653301857515644791584016, 18.06806473198259279694713731214, 18.92555239269327346819400625581, 19.63451224045291473752077521000, 20.756267329167682954569440217645, 20.93004423978071433939703280929, 22.42917945195613580512557323214

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