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Properties

Label 1-744-744.77-r0-0-0
Degree $1$
Conductor $744$
Sign $0.569 + 0.822i$
Analytic cond. $3.45512$
Root an. cond. $3.45512$
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  + 5-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 + 0.587i)23-s + 25-s + (−0.309 + 0.951i)29-s + (−0.809 − 0.587i)35-s + 37-s + (−0.309 + 0.951i)41-s + (0.309 − 0.951i)43-s + (−0.309 − 0.951i)47-s + (0.309 + 0.951i)49-s + ⋯
L(s)  = 1  + 5-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 + 0.587i)23-s + 25-s + (−0.309 + 0.951i)29-s + (−0.809 − 0.587i)35-s + 37-s + (−0.309 + 0.951i)41-s + (0.309 − 0.951i)43-s + (−0.309 − 0.951i)47-s + (0.309 + 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(744\)    =    \(2^{3} \cdot 3 \cdot 31\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(3.45512\)
Root analytic conductor: \(3.45512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{744} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 744,\ (0:\ ),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.318888313 + 0.6911552819i\)
\(L(\frac12)\) \(\approx\) \(1.318888313 + 0.6911552819i\)
\(L(1)\) \(\approx\) \(1.149291832 + 0.1792735651i\)
\(L(1)\) \(\approx\) \(1.149291832 + 0.1792735651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
31 \( 1 \)
good5 \( 1 + T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−22.22996341800126507404199822682, −21.78215779864518050150059369252, −20.74546939519834658270894444104, −19.929187725389351655752916454451, −19.124608596886659054234931598310, −18.182819075333614668561852371368, −17.59282966769135952803324599703, −16.665060251634498781343680187425, −15.8535407411042010019869431683, −15.03612733471058608163366137432, −13.99966224412436503111086473194, −13.286679618386331657415509692568, −12.645369060947552837430695475193, −11.53407865542911967933696494295, −10.66332683248831974111454499737, −9.627746393275013768171527913845, −9.117608221383329722053049445550, −8.1970594609604545403265253258, −6.73920651322857655963733378072, −6.18267158757104190264074268516, −5.411320790250733396148487437648, −4.16694802727929502618443393578, −2.92229125701930793101166168279, −2.24354505231881746382291584342, −0.71851141415572785459443525998, 1.40783035423471033203616192399, 2.17309124594149508020311115013, 3.63706307569315755427843953141, 4.3144001761996579210658338100, 5.68188827113903351794801635780, 6.510161481526949935289742501339, 7.03327567487988909639751317406, 8.46867687115212713091213061523, 9.39054289684195853689864745298, 9.94656098684370927976917579647, 10.81733727645564681219751831606, 11.92511341545803395405118839294, 12.85309311690849587639093479838, 13.5613377685694273217913560051, 14.29155366084805775985271497018, 15.12338129913376547821131556560, 16.473002725003026943276572056924, 16.73015257721611143699745414574, 17.723953567213694469209343411759, 18.45284774742467076147751539671, 19.52596317404460300448145312544, 20.087652290576709939526981005334, 21.04332763724589517202963185667, 21.88803437214423908095740890597, 22.42859397077954901460574247574

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