↑

Properties

Label 1-4719-4719.2804-r0-0-0
Degree $1$
Conductor $4719$
Sign $-0.265 + 0.964i$
Analytic cond. $21.9149$
Root an. cond. $21.9149$
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  + (0.981 + 0.189i)2-s + (0.928 + 0.371i)4-s + (−0.841 + 0.540i)5-s + (−0.723 + 0.690i)7-s + (0.841 + 0.540i)8-s + (−0.928 + 0.371i)10-s + (−0.841 + 0.540i)14-s + (0.723 + 0.690i)16-s + (0.580 + 0.814i)17-s + (0.995 + 0.0950i)19-s + (−0.981 + 0.189i)20-s + (−0.723 − 0.690i)23-s + (0.415 − 0.909i)25-s + (−0.928 + 0.371i)28-s + (0.580 − 0.814i)29-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)2-s + (0.928 + 0.371i)4-s + (−0.841 + 0.540i)5-s + (−0.723 + 0.690i)7-s + (0.841 + 0.540i)8-s + (−0.928 + 0.371i)10-s + (−0.841 + 0.540i)14-s + (0.723 + 0.690i)16-s + (0.580 + 0.814i)17-s + (0.995 + 0.0950i)19-s + (−0.981 + 0.189i)20-s + (−0.723 − 0.690i)23-s + (0.415 − 0.909i)25-s + (−0.928 + 0.371i)28-s + (0.580 − 0.814i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4719\)    =    \(3 \cdot 11^{2} \cdot 13\)
Sign: $-0.265 + 0.964i$
Analytic conductor: \(21.9149\)
Root analytic conductor: \(21.9149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4719} (2804, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4719,\ (0:\ ),\ -0.265 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.745189776 + 2.289911813i\)
\(L(\frac12)\) \(\approx\) \(1.745189776 + 2.289911813i\)
\(L(1)\) \(\approx\) \(1.552272141 + 0.6909567243i\)
\(L(1)\) \(\approx\) \(1.552272141 + 0.6909567243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.981 + 0.189i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.723 + 0.690i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
23 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (0.580 - 0.814i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (0.928 - 0.371i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (-0.981 + 0.189i)T \)
61 \( 1 + (-0.981 + 0.189i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.995 + 0.0950i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.995 - 0.0950i)T \)
97 \( 1 + (-0.888 + 0.458i)T \)
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

−18.08609996973269959753333935085, −16.84798838453932485786730802690, −16.53121267616487223603609682000, −15.77903726299077508522227376349, −15.50053183283621953965865316765, −14.42969104409127315259240572225, −13.87704074505044538532496112136, −13.27780360691993187751567511918, −12.53240269453524654836749451465, −12.05100011768472386580198116729, −11.35817043898808885398266244884, −10.75860829414696588504593662476, −9.743299692153652994251817111093, −9.401117002709163846742874108296, −8.069390584798394442175023546750, −7.46435823821850531289111623154, −7.00431057380764439393801024803, −5.99059068610931249067225787189, −5.35032620716099844260817037693, −4.52565748570763518017562828096, −3.91208015789486616829507028305, −3.288997799926776881501461256552, −2.58932512869295120297244037822, −1.291510837519578464193455379608, −0.63261016282315614939794269775, 1.054387482718632009909827644026, 2.42399410158185633194923675999, 2.806696320486772118889264127872, 3.71472267523646335203235560668, 4.143848157147567640883105687973, 5.150847260494012757417527444727, 6.01155525814050928931991199744, 6.35168894424551146638380500755, 7.31409386114940795389127476843, 7.837484415156966860480475097196, 8.571235583646678842817409136, 9.60126151285011227554184860284, 10.44390681665465265849267279002, 11.02744795511400671466100195846, 11.926474397507749815905978733877, 12.29995852822410005924456846159, 12.78719097908198940659658793140, 13.86395856335684752398342306270, 14.30971056287839923984315886824, 15.04662726571398733544715318547, 15.65191453586726993075598037929, 16.07908241447706427029754990881, 16.68341380298503250649640298418, 17.67549565371766417021397940213, 18.50815144665386786278398213289

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