Properties

Label 1-3024-3024.1235-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

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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: $\chi_{3024} (1235, \cdot )$
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 \( 1 + (0.939 + 0.342i)T \)
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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

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

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