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Properties

Label 1-4004-4004.691-r1-0-0
Degree $1$
Conductor $4004$
Sign $0.501 + 0.864i$
Analytic cond. $430.289$
Root an. cond. $430.289$
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  + (−0.978 + 0.207i)3-s + (−0.406 + 0.913i)5-s + (0.913 − 0.406i)9-s + (0.207 − 0.978i)15-s + (−0.809 + 0.587i)17-s + (0.743 − 0.669i)19-s + 23-s + (−0.669 − 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (−0.406 − 0.913i)31-s + (−0.951 + 0.309i)37-s + (−0.743 + 0.669i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)3-s + (−0.406 + 0.913i)5-s + (0.913 − 0.406i)9-s + (0.207 − 0.978i)15-s + (−0.809 + 0.587i)17-s + (0.743 − 0.669i)19-s + 23-s + (−0.669 − 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (−0.406 − 0.913i)31-s + (−0.951 + 0.309i)37-s + (−0.743 + 0.669i)41-s + (−0.5 − 0.866i)43-s + i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.501 + 0.864i$
Analytic conductor: \(430.289\)
Root analytic conductor: \(430.289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4004,\ (1:\ ),\ 0.501 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6577229683 + 0.3788333139i\)
\(L(\frac12)\) \(\approx\) \(0.6577229683 + 0.3788333139i\)
\(L(1)\) \(\approx\) \(0.6386876550 + 0.1265363318i\)
\(L(1)\) \(\approx\) \(0.6386876550 + 0.1265363318i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.406 - 0.913i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.207 - 0.978i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (-0.207 + 0.978i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.994 - 0.104i)T \)
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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

−18.07506030493533444840226884923, −17.50110174499741474189136875782, −16.74462557122905935284385648666, −16.30897156074433167964955725287, −15.68513016589850645594178839202, −15.01868654579999629560011892538, −13.87102259156746381892936584361, −13.31345808228712919424715387704, −12.4584517521180023915448231863, −12.214358571889367570248392124751, −11.2401648759942260486158203466, −10.931762814653363727985825097357, −9.85463365663189841837153507990, −9.20586181474616344372291238840, −8.46069863300746542178844618555, −7.51558993888096155083488907696, −7.05085056778097517723790739256, −6.14015039479824837297154135684, −5.19610846600967888268653305985, −4.99169495962678970077798873395, −4.03473684107496321757878558444, −3.20934222180422718083257335157, −1.85096781350817707630152619152, −1.23419524443990530913448737814, −0.3002840916154499352176269213, 0.40046915803956492429835377942, 1.54664671959787294878359788057, 2.51973130061868730587761569742, 3.52899485474366688019706881418, 4.080945822308012473684020971377, 5.05017302949373991622628896182, 5.65268547601633602953636856541, 6.658437302656198202266374952266, 6.94355995263074083568442016886, 7.73704224644533015780721367300, 8.76695873828503967177347943122, 9.56298517087909534290791658576, 10.38118552190957004633505199552, 10.84934498847393470760187119158, 11.60895853329603201404331970842, 11.87049711055820511006320469485, 13.086970317655353098897660844926, 13.39761602673385603366066295605, 14.559689359826668883526027288762, 15.32841628549156244621855124432, 15.450331876869577328320608064243, 16.490600214301315645848611520741, 17.09570098692483183275114900080, 17.71230164812540993564035161158, 18.487927358025860211900453916690

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