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Properties

Label 1-1001-1001.500-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.929 + 0.367i$
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  + (0.587 + 0.809i)2-s + (0.978 + 0.207i)3-s + (−0.309 + 0.951i)4-s + (0.406 + 0.913i)5-s + (0.406 + 0.913i)6-s + (−0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.207 + 0.978i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.207 + 0.978i)18-s + (0.743 + 0.669i)19-s + (−0.994 + 0.104i)20-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.978 + 0.207i)3-s + (−0.309 + 0.951i)4-s + (0.406 + 0.913i)5-s + (0.406 + 0.913i)6-s + (−0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.207 + 0.978i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (0.207 + 0.978i)18-s + (0.743 + 0.669i)19-s + (−0.994 + 0.104i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.929 + 0.367i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.929 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5191601451 + 2.725153530i\)
\(L(\frac12)\) \(\approx\) \(0.5191601451 + 2.725153530i\)
\(L(1)\) \(\approx\) \(1.284911414 + 1.399102650i\)
\(L(1)\) \(\approx\) \(1.284911414 + 1.399102650i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
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

−21.26093513826042995796520955469, −20.36801114725856389170546058497, −20.007783606160019104625696455, −19.35191615680949716990637716406, −18.28659318561691896058580654486, −17.74429458930431235467463532584, −16.44496557341681542060133211859, −15.522940316990631646609281837423, −14.81608935298087608416968061849, −13.885531784224163880176392840425, −13.30736012798357749488249787945, −12.78806669428746112275763878407, −11.92782863263236446977746153394, −10.95498246191092011965298061521, −9.8144530699612805139411291092, −9.33525957567181611413435707723, −8.57710541180888561250193624311, −7.56065874105210299046683822343, −6.30226509306224109115469275990, −5.443320685108667573206746980292, −4.33313316606181448092396203148, −3.81083047916132165378660677437, −2.48912442245214548317304853646, −1.93214020569463278265593682364, −0.82688721378269184213019766728, 1.93024212040212512544581092381, 2.85470929288931759512218244651, 3.58823452951212343933497317760, 4.479875524580446360314150922782, 5.58312680668739360425486465622, 6.48719278575276635588736306423, 7.37863778305122845416698629463, 7.89544193301791825115382505820, 9.05098956341653662260921879235, 9.64221114566851001856064115106, 10.70558650085901535352149631621, 11.74012312131220277222655236133, 12.85079473449382366225265961941, 13.572481290239147698226205467079, 14.29369491447664048559455804969, 14.63864293223735769458145305529, 15.66951989925289825066268462689, 16.09633011803819510522250115694, 17.23369458993623032790583873320, 18.262505517172216516194041880824, 18.52871710807198369682987165535, 19.85659815355789181408025990673, 20.532055890477335522294511252616, 21.43634964493961031008318549100, 22.06095847919448466424618962807

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