↑

Properties

Label 1-3024-3024.1643-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.998 - 0.0622i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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  + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (0.642 + 0.766i)13-s + 17-s − i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)11-s + (0.642 + 0.766i)13-s + 17-s − i·19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.998 - 0.0622i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.998 - 0.0622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01098550465 + 0.3523981556i\)
\(L(\frac12)\) \(\approx\) \(0.01098550465 + 0.3523981556i\)
\(L(1)\) \(\approx\) \(0.8737025227 + 0.1031184774i\)
\(L(1)\) \(\approx\) \(0.8737025227 + 0.1031184774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (0.984 + 0.173i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (-0.342 - 0.939i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.642 + 0.766i)T \)
89 \( 1 - T \)
97 \( 1 + (0.939 - 0.342i)T \)
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

−18.726694933626345219579034662752, −17.904102616418444952029305521493, −16.82128862431418959556488477609, −16.59027195833840132149585238658, −15.694401428926870883177786526721, −15.07824119286318308781708117510, −14.366795719415876714103350823270, −13.69877130560888547470152209680, −12.67519343203184894978068284942, −12.08603590907079991719097846891, −11.65829829952108173896757265735, −10.61516809671405173023124086170, −10.16505560909408919165565986407, −8.98712758838361423404726551786, −8.50102004731227215563591718388, −7.706931084880719801936621319109, −7.11889576862682160294826774142, −6.01310548010654959886652787452, −5.52299268409193273390299485054, −4.3315759613244311808943219423, −3.7135337662651218783410305319, −3.20101021597475758066613374345, −1.82642222116035798910068237129, −0.97588784711404779030218090032, −0.067003325919528941235541465638, 1.07672723559089282534878186311, 1.87250219439015125404321070086, 3.15383393630349874282012404760, 3.762407310802561868195564263013, 4.37387645884797361888342181368, 5.36878567155564833962448915651, 6.27776274147255096087406089328, 7.09581564963331486893959873512, 7.5620284133753672606953526742, 8.57749682580757949031360259749, 9.125136055918669185541270724561, 9.919656448059382658351904061678, 11.01221490078192923794808772686, 11.387594056308431957109981848949, 12.114310912633115389981833577728, 12.72630081841879091261375987885, 13.80128245688741663075525257730, 14.32030048812507060932791015854, 15.08781608967093142616626936102, 15.71019533436064245686115737991, 16.50429758872876250252920859887, 16.92659341971705161916284035282, 17.98660680452562004040187149921, 18.59030726732877733564636482999, 19.27215992034824186311863851747

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