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Properties

Label 1-1001-1001.200-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.742 + 0.669i$
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  + (0.406 + 0.913i)2-s + (−0.978 + 0.207i)3-s + (−0.669 + 0.743i)4-s + (−0.994 + 0.104i)5-s + (−0.587 − 0.809i)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 + (0.951 − 0.309i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.743 + 0.669i)18-s + (−0.743 + 0.669i)19-s + (0.587 − 0.809i)20-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.978 + 0.207i)3-s + (−0.669 + 0.743i)4-s + (−0.994 + 0.104i)5-s + (−0.587 − 0.809i)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 + (0.951 − 0.309i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (0.743 + 0.669i)18-s + (−0.743 + 0.669i)19-s + (0.587 − 0.809i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.742 + 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2947640437 + 0.7669704917i\)
\(L(\frac12)\) \(\approx\) \(0.2947640437 + 0.7669704917i\)
\(L(1)\) \(\approx\) \(0.5979166070 + 0.4592611733i\)
\(L(1)\) \(\approx\) \(0.5979166070 + 0.4592611733i\)

Euler product

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

−21.33698494627021890113660284861, −20.73752343707350732646572077121, −19.60796876043859372109683543918, −19.16668853958590378339287461574, −18.426631166047725864924732279198, −17.572589858814787100053078411299, −16.71736265065516557978437001, −15.76406455902207714963616437202, −15.082793013229376486058488746338, −14.06679180059246201594448623174, −12.98288214809540838223820828531, −12.53857669578980286974549626735, −11.591434112655342629027237082817, −11.2791177215457738981776335159, −10.3845026186165378290249102662, −9.50240456103188119622692056313, −8.40528922527293963722275218117, −7.367969111357101677500292051975, −6.44398546947464798307400863355, −5.33476796292202526774690316295, −4.73179485588692894327435711621, −3.83052394648840221371395748261, −2.864671964566626565665412128003, −1.48567207986255999073289571748, −0.5228877287255581109325174132, 0.8689771672325103421966255882, 2.89041553110263655951815510692, 4.16226269957745094013117019124, 4.35195627166547533721383945322, 5.63578042712941383864811122505, 6.21786409304547017239858699447, 7.20189674893345885430486031538, 7.868477049136714835160222432856, 8.77446051417302812098876508023, 9.93445310169361309312236969969, 10.84075058758934430653558127506, 11.778898084995309496049124746915, 12.43942859244502262922466984924, 13.01665230747599026672857422490, 14.405204060877636786681251364620, 14.93669699080671931978885460522, 15.79040986343276825943923625388, 16.38262923745884241180794752859, 17.01170755109796832572834453231, 17.79509778729419828930314324257, 18.761605559429653492342388478352, 19.26431425068589909195165701618, 20.84946975541895241394244182483, 21.24114712845390257027968635499, 22.33078363698238078288116792060

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