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Properties

Label 2-1001-7.2-c1-0-15
Degree $2$
Conductor $1001$
Sign $0.781 - 0.623i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.865 − 1.49i)2-s + (0.0316 + 0.0548i)3-s + (−0.498 − 0.863i)4-s + (−1.64 + 2.84i)5-s + 0.109·6-s + (−2.63 + 0.229i)7-s + 1.73·8-s + (1.49 − 2.59i)9-s + (2.83 + 4.91i)10-s + (−0.5 − 0.866i)11-s + (0.0315 − 0.0547i)12-s − 13-s + (−1.93 + 4.15i)14-s − 0.207·15-s + (2.49 − 4.33i)16-s + (3.93 + 6.82i)17-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (0.0182 + 0.0316i)3-s + (−0.249 − 0.431i)4-s + (−0.733 + 1.27i)5-s + 0.0448·6-s + (−0.996 + 0.0867i)7-s + 0.614·8-s + (0.499 − 0.864i)9-s + (0.897 + 1.55i)10-s + (−0.150 − 0.261i)11-s + (0.00912 − 0.0157i)12-s − 0.277·13-s + (−0.517 + 1.10i)14-s − 0.0536·15-s + (0.624 − 1.08i)16-s + (0.955 + 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.781 - 0.623i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ 0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.565938163\)
\(L(\frac12)\) \(\approx\) \(1.565938163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 - 0.229i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.865 + 1.49i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0316 - 0.0548i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.64 - 2.84i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (-3.93 - 6.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.87 - 6.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.05 - 7.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 + (-2.04 - 3.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.05 - 1.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.97T + 41T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 + (2.32 - 4.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.99 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.78 + 8.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.80 + 8.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.40 - 2.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.59T + 71T^{2} \)
73 \( 1 + (5.23 + 9.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.20 - 5.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (-6.65 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.22877670915801210386123269394, −9.838340765980913251142756229631, −8.279920970086883707689975435172, −7.53124283127252016526993016832, −6.55021132967162504154157043255, −5.87913415302228542281106139345, −4.16859820774808880323583380629, −3.50185469425703193846683969243, −3.15650485586946916241837415356, −1.66000982664686018941979689105, 0.59199933157526662562040960136, 2.53557712055325033172981951320, 4.18259630005144889561742340400, 4.73177183508161931327180407396, 5.36824841534828449554756102042, 6.63940043536558412272447011702, 7.19518454425066577685395173679, 8.046132055144916360821092339977, 8.760543188590961360876398861762, 9.862119014202479200990233316946

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