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Properties

Label 1-1001-1001.996-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: $\chi_{1001} (996, \cdot )$
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

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

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