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Properties

Label 1-1001-1001.184-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.404 + 0.914i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (0.951 + 0.309i)2-s + (0.913 + 0.406i)3-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (0.743 + 0.669i)6-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.406 − 0.913i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.406 + 0.913i)18-s + (0.994 − 0.104i)19-s + (−0.207 − 0.978i)20-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.913 + 0.406i)3-s + (0.809 + 0.587i)4-s + (−0.743 − 0.669i)5-s + (0.743 + 0.669i)6-s + (0.587 + 0.809i)8-s + (0.669 + 0.743i)9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.406 − 0.913i)15-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.406 + 0.913i)18-s + (0.994 − 0.104i)19-s + (−0.207 − 0.978i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.404 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.939368032 + 1.913864343i\)
\(L(\frac12)\) \(\approx\) \(2.939368032 + 1.913864343i\)
\(L(1)\) \(\approx\) \(2.151892694 + 0.7854445733i\)
\(L(1)\) \(\approx\) \(2.151892694 + 0.7854445733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (0.743 - 0.669i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.994 - 0.104i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−21.4612599717144066955966389788, −20.64527155049196329299510895622, −19.984102152037250464221585606241, −19.43502044184551818614213175146, −18.57222380914680673871791185212, −18.04170061604930847539911545781, −16.32053585420136949442841880535, −15.77100688517870030409198195521, −14.917890894960685024759981366748, −14.34475101482322193419964106836, −13.68032797218612216363842506886, −12.89643349277221116379375362867, −11.76636872277866539715356738107, −11.623713118740073452986855965475, −10.21890728194204039259549968627, −9.6403691473486237552852001521, −8.26004914057004562423969731607, −7.466554652110772214593235046519, −6.87702679674825765816620097441, −5.87314391649031520212025248961, −4.63302703179921893049373330588, −3.74023669369053378642674361931, −3.050147982846288402896685094996, −2.31514870922204092848372168864, −1.05303640375144155765419696716, 1.5081750092516738845789014242, 2.62425603683367616331909699149, 3.70585348483540051460980278782, 4.096672587080992754529853139550, 5.07823182755431212032555309970, 5.95816167386786390753589523300, 7.35356842278115304531622847453, 7.83648410628986066633597371881, 8.61134230050704535058618297683, 9.59058396084164303937868918079, 10.68779912468943775401361464417, 11.61740172994452213150307987417, 12.457020785190902403248267941980, 13.15236770581676738018149731076, 13.93207337178200175319960790258, 14.77464613335765815176323179384, 15.343268303513799211817940274141, 16.17752188003194513863891081782, 16.55363260022453267317790693096, 17.73715102772952528263342729679, 19.08452317486856150001480973194, 19.69745746794112241590383407725, 20.47576829335538326165330738703, 20.86549006508144446358410003075, 21.861100616315698992089243411514

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