↑

Properties

Label 1-709-709.144-r0-0-0
Degree $1$
Conductor $709$
Sign $0.460 - 0.887i$
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  + (−0.998 + 0.0532i)2-s + (−0.964 − 0.263i)3-s + (0.994 − 0.106i)4-s + (−0.132 + 0.991i)5-s + (0.977 + 0.211i)6-s + (−0.931 − 0.364i)7-s + (−0.987 + 0.159i)8-s + (0.861 + 0.507i)9-s + (0.0797 − 0.996i)10-s + (−0.437 + 0.899i)11-s + (−0.987 − 0.159i)12-s + (−0.931 + 0.364i)13-s + (0.949 + 0.314i)14-s + (0.388 − 0.921i)15-s + (0.977 − 0.211i)16-s + (−0.437 − 0.899i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0532i)2-s + (−0.964 − 0.263i)3-s + (0.994 − 0.106i)4-s + (−0.132 + 0.991i)5-s + (0.977 + 0.211i)6-s + (−0.931 − 0.364i)7-s + (−0.987 + 0.159i)8-s + (0.861 + 0.507i)9-s + (0.0797 − 0.996i)10-s + (−0.437 + 0.899i)11-s + (−0.987 − 0.159i)12-s + (−0.931 + 0.364i)13-s + (0.949 + 0.314i)14-s + (0.388 − 0.921i)15-s + (0.977 − 0.211i)16-s + (−0.437 − 0.899i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1530323352 - 0.09306520167i\)
\(L(\frac12)\) \(\approx\) \(0.1530323352 - 0.09306520167i\)
\(L(1)\) \(\approx\) \(0.3552490356 + 0.04531200935i\)
\(L(1)\) \(\approx\) \(0.3552490356 + 0.04531200935i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.998 + 0.0532i)T \)
3 \( 1 + (-0.964 - 0.263i)T \)
5 \( 1 + (-0.132 + 0.991i)T \)
7 \( 1 + (-0.931 - 0.364i)T \)
11 \( 1 + (-0.437 + 0.899i)T \)
13 \( 1 + (-0.931 + 0.364i)T \)
17 \( 1 + (-0.437 - 0.899i)T \)
19 \( 1 + (-0.987 + 0.159i)T \)
23 \( 1 + (0.484 + 0.874i)T \)
29 \( 1 + (-0.887 + 0.461i)T \)
31 \( 1 + (0.388 - 0.921i)T \)
37 \( 1 + (-0.931 - 0.364i)T \)
41 \( 1 + (0.574 - 0.818i)T \)
43 \( 1 + (0.658 + 0.752i)T \)
47 \( 1 + (0.484 + 0.874i)T \)
53 \( 1 + (0.388 - 0.921i)T \)
59 \( 1 + (-0.987 + 0.159i)T \)
61 \( 1 + (-0.617 - 0.786i)T \)
67 \( 1 + (0.185 + 0.982i)T \)
71 \( 1 + (0.0797 + 0.996i)T \)
73 \( 1 + (-0.530 + 0.847i)T \)
79 \( 1 + (-0.833 + 0.552i)T \)
83 \( 1 + (0.734 - 0.678i)T \)
89 \( 1 + (0.910 + 0.413i)T \)
97 \( 1 + (-0.0266 - 0.999i)T \)
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

−22.732925112694662362438984822282, −21.66307884014302389962932554593, −21.25410801080247649076774842573, −20.17084756526233834759317136423, −19.31362237023652195119833658594, −18.77638232851832529909979788526, −17.65814899212801972469497945173, −16.85657245460064333209923039498, −16.60204635772634535833557852049, −15.59877663922186236269460017943, −15.15029659243491117068911755106, −13.2076199098870141130777330437, −12.49759402708738862181042924264, −11.985725267844958653366334147, −10.752407302997786873353200443690, −10.31522340309594903726543746179, −9.188554357735542391688053681210, −8.65902705534397078993830146234, −7.51983036966551698347343200257, −6.385892411887841903059838827813, −5.803201748940496058140605250822, −4.71812147529370466074976275347, −3.444544639461362117997014219305, −2.147919520567352941825329874566, −0.70936727847224950063041597776, 0.20389955376803010024620908180, 1.92569424504720150231685327361, 2.76741112633560571883466242197, 4.19265078263927764422625132830, 5.56647560486968977983070002023, 6.53665430631588497833467839713, 7.22595330078635379255092243173, 7.532812577396301245604037408034, 9.3189380686421834716526715804, 9.93484874978354480877621890956, 10.683528134420259763568404386753, 11.38163482416838437931625526105, 12.2923253219826958218209857117, 13.08737770063434527041969270515, 14.4390138100799601716903191185, 15.476427090746660642385905977265, 15.96414039960326116928056980342, 17.117728093877354819264632726532, 17.4431834713414807051135431330, 18.4335192413150979076351425508, 19.05317340536003545486783664327, 19.62093206779378468981728845534, 20.750248186199173750946363218201, 21.773626379321043363468652957866, 22.64173780130878282976462606129

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