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Properties

Label 1-1001-1001.139-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.982 + 0.184i$
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  + (−0.669 + 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.309 + 0.951i)5-s + (0.669 + 0.743i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.809 − 0.587i)18-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 − 0.994i)4-s + (−0.309 + 0.951i)5-s + (0.669 + 0.743i)6-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s − 12-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (0.669 + 0.743i)17-s + (0.809 − 0.587i)18-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.982 + 0.184i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.982 + 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9080611287 + 0.08462713731i\)
\(L(\frac12)\) \(\approx\) \(0.9080611287 + 0.08462713731i\)
\(L(1)\) \(\approx\) \(0.7444542059 + 0.07661239685i\)
\(L(1)\) \(\approx\) \(0.7444542059 + 0.07661239685i\)

Euler product

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

−21.336041794607447641502937514631, −20.76085466005080178560766694226, −20.13150344401094607726997498684, −19.6121095268785299582819792390, −18.57601083128718639855817124889, −17.65807728359942539350790894762, −16.87580725732195883630702938943, −16.13000037760231560802264798385, −15.83447476490171539007314729462, −14.478938988915967871183240101568, −13.62284739351119363238683712393, −12.62774918029489249065733425615, −11.803817930115744038772240287888, −11.26138419589760458605691387613, −10.19760337934258238379182967405, −9.49135307789550893140288651638, −8.993439084868456225303376236342, −8.03766212289522786604392285111, −7.37093096517356423753207451974, −5.60415863641205185820003134626, −4.91197866709319335172377265547, −3.828906417364485516789317799836, −3.31737368959522368882465864052, −2.0028619496033129898574307851, −0.77138954204028631583063542655, 0.74737113688057975213319603098, 1.96764946065432647931946519967, 2.901533974670653333760719188489, 4.2061042120758387502178996836, 5.753531505729547820766053149813, 6.144460687853147470447851582989, 7.23195796631045108117973699930, 7.61241950939180982584656749296, 8.41580411137737436194498650848, 9.43962346577794136133859248497, 10.362465080999270537727402380961, 11.227184371039289123904537111573, 11.96057868130116362886843604483, 13.154537453815061438862484856176, 13.946554081326851491304302086329, 14.68974716768129219289082736696, 15.20266635092802955868101224205, 16.32310318158780460272715032235, 17.103913589232505162114445140217, 17.907026589960875270983867510541, 18.62225659280433733994511638419, 18.92218178517023450294274297547, 19.856259538199950770269676228464, 20.47624910133889260132604559560, 22.037067740864134344399339174322

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