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Properties

Label 1-1001-1001.180-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.784 - 0.619i$
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 + (0.104 − 0.994i)3-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.207 + 0.978i)6-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.994 − 0.104i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.994 − 0.104i)18-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.104 − 0.994i)3-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.207 + 0.978i)6-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.994 − 0.104i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.994 − 0.104i)18-s + (−0.406 − 0.913i)19-s + (0.743 + 0.669i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.784 - 0.619i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.784 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1387733677 - 0.3994871404i\)
\(L(\frac12)\) \(\approx\) \(0.1387733677 - 0.3994871404i\)
\(L(1)\) \(\approx\) \(0.5968589575 - 0.1209941309i\)
\(L(1)\) \(\approx\) \(0.5968589575 - 0.1209941309i\)

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 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.207 + 0.978i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−21.47194744442141383446815953526, −21.222754515759571638872499528638, −20.191085865175126485742827940, −19.957027019338408156050108336743, −18.92034863436182096067743212086, −17.93479062023400643813740585718, −17.052473487738172660800450640118, −16.60625899727333640955117649405, −15.981847343902983103343281940396, −15.10044592406313324390422147307, −14.22339496313027811706363759326, −12.95332388810889706595583468111, −12.27670181886807311889604807781, −11.37077024685103182107053534014, −10.47536033780780868471396038629, −9.83974874477442385452181727006, −9.14159320546435194480819108171, −8.335355930415799434833027395673, −7.81100135710875021909409228229, −6.232253802172834394109371475366, −5.54322839866058499019570962165, −4.2518700943093757682998005432, −3.630755685971442039463229128077, −2.33645153405199079683724960342, −1.36441380873407214397013553742, 0.23697019490768141291989861849, 1.63550437043673812494153632222, 2.46700878368160036906075699743, 3.24116407090365078678730442241, 5.08814395799822053753454257336, 6.24972776249232517056156582340, 6.61215263924761803053556258892, 7.550271720253551088050954408503, 8.0820271945617716842547785681, 9.19904485813187779411471992711, 9.91661711912270650607171178360, 10.9876637470602961682364524479, 11.51455007010971646639919044431, 12.42847843769651637836260455121, 13.69095724704952416141018285588, 14.192505151540278235754279062438, 15.071576612211863406656214265435, 15.8653174860852792384936741566, 16.93534244712015031623902456606, 17.69815518532257631622320349102, 18.19139485654280573497337126227, 18.85549573431556218361019094499, 19.51489984528258660715749376728, 20.20722948024117961076664809699, 21.18472232382817156370964738914

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