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Properties

Label 1-1001-1001.173-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.999 - 0.0443i$
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 + (0.809 − 0.587i)3-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (0.978 + 0.207i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 + 0.866i)12-s + (0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.913 − 0.406i)18-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)20-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.809 − 0.587i)3-s + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (0.978 + 0.207i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 10-s + (0.5 + 0.866i)12-s + (0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (0.669 − 0.743i)17-s + (0.913 − 0.406i)18-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.999 - 0.0443i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.999 - 0.0443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.198273200 - 0.07100242597i\)
\(L(\frac12)\) \(\approx\) \(3.198273200 - 0.07100242597i\)
\(L(1)\) \(\approx\) \(2.088811121 + 0.1756538539i\)
\(L(1)\) \(\approx\) \(2.088811121 + 0.1756538539i\)

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 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−21.58254833381777549445379142962, −20.94459684113921954345510730936, −20.32989548866297864828649445229, −19.44545086140648193182913818766, −18.74410757523805452560143963809, −18.1174545058203268864499446132, −16.86107572621325037014886008903, −15.864719678888226577334019403499, −15.00149865891093226460774734076, −14.36321705106710427070057575853, −13.93191506580022105808866504202, −13.057434939293167581129374763121, −12.1673347723510529718328644915, −11.0748276271270314895912667932, −10.28021161084378513355473605519, −9.92426333873582329992776432065, −8.99008796397231023789328771193, −7.9270521246737776740384449642, −6.74089268828888294719415096820, −5.77820130457385630461461135666, −4.96490564808234793126212711403, −3.82534890829917596187521548728, −3.223899549174580253070402969347, −2.33058983286979688808884679668, −1.50154445635612905795240873963, 1.06536953760258472461639481414, 2.315319440727483895348533276558, 3.16784403740670525966812630910, 4.20795690619283350713274012249, 5.241398643623687051320259405227, 5.94653300882133459181837303597, 7.01563455388236023089225170644, 7.64647986983282380487083878446, 8.59692020056767477928007571330, 9.19656482050245108929697131642, 10.08670065849780767290362000293, 11.978855102542010159750690656605, 12.07892544880818821249977104296, 13.365280007872796594002385881153, 13.670798151975002384955613266784, 14.23649551168630173052755005690, 15.40152150571401103186837524098, 15.89044683888799168235019864099, 16.97031276612438466468732469527, 17.63569915093383197571286473513, 18.31286330079685032160900237543, 19.38593343471865305926842012124, 20.385219485553021521646548906436, 20.84477554359793700961885440066, 21.58949328274143946772083951121

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