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Properties

Label 1-1001-1001.128-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.0324 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0324 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0324 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.0324 - 0.999i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.0324 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512949317 - 1.464625863i\)
\(L(\frac12)\) \(\approx\) \(1.512949317 - 1.464625863i\)
\(L(1)\) \(\approx\) \(1.532980231 - 0.4340723247i\)
\(L(1)\) \(\approx\) \(1.532980231 - 0.4340723247i\)

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

−21.96485947924932647599400569818, −21.36804138643723598623194489572, −20.21348503013870328713873719387, −19.49835203641947670972209719162, −18.78880732204118500325826801306, −17.89056603177126024365366866907, −17.16597712560108961788291824169, −16.27040578212206208181678124340, −15.30799925198750889041607372271, −14.45132354621641886580723251605, −14.10665897452148125391437384369, −13.10706721479541100979631453730, −12.38844379532115193697616547621, −11.6958522205557549624240431372, −10.92609264469497811717059615221, −10.05187812430918156612454757610, −8.26947696427137293094692540983, −7.884983279733164786537031254969, −6.93258181630712071035463900619, −6.20052356668016060206537344967, −5.71204303172063004412921183648, −4.27611680086211412850925245767, −3.39786002888445415255544507865, −2.47477691446541938245162628834, −1.62391789846468943336881279636, 0.61985305927069474822326293488, 2.102317633965440993352953845194, 3.18910946750290692163992313440, 4.08474363049389126909945050752, 4.795376805652404977717519368392, 5.38650088749075649129424315203, 6.3252287996275079410918428672, 7.54350646573639651460961792638, 8.67825956132929125437406166632, 9.513675504926506654072882507906, 10.26769978887863169579511227321, 11.26314316884118124010258445887, 11.8473676937719016693556192375, 12.62979559064139931180981041934, 13.62153452933595748959375591337, 14.23524224650033107167892322111, 15.378941542042763021031981912, 15.68430191170113947478031213573, 16.57430791873526401315184426928, 17.13850410690335574182207321772, 18.4553716274831666116581850116, 19.602987739498713313457274999469, 20.27606678143665481930588503057, 20.641834298418640816592827335128, 21.53359559214804088191637762112

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