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Properties

Label 2-1001-7.4-c1-0-74
Degree $2$
Conductor $1001$
Sign $-0.991 + 0.126i$
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  + (0.0667 + 0.115i)2-s + (1.25 − 2.17i)3-s + (0.991 − 1.71i)4-s + (−1.25 − 2.17i)5-s + 0.335·6-s + (−2 + 1.73i)7-s + 0.531·8-s + (−1.65 − 2.87i)9-s + (0.167 − 0.290i)10-s + (0.5 − 0.866i)11-s + (−2.49 − 4.31i)12-s − 13-s + (−0.333 − 0.115i)14-s − 6.31·15-s + (−1.94 − 3.37i)16-s + (−1.23 + 2.13i)17-s + ⋯
L(s)  = 1  + (0.0471 + 0.0816i)2-s + (0.725 − 1.25i)3-s + (0.495 − 0.858i)4-s + (−0.562 − 0.973i)5-s + 0.136·6-s + (−0.755 + 0.654i)7-s + 0.187·8-s + (−0.552 − 0.957i)9-s + (0.0530 − 0.0918i)10-s + (0.150 − 0.261i)11-s + (−0.719 − 1.24i)12-s − 0.277·13-s + (−0.0891 − 0.0308i)14-s − 1.63·15-s + (−0.486 − 0.842i)16-s + (−0.299 + 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628784017\)
\(L(\frac12)\) \(\approx\) \(1.628784017\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 - 1.73i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.0667 - 0.115i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.25 + 2.17i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (1.23 - 2.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.03 - 1.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.55 + 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 + (-2.74 + 4.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.60 - 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.397T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 + (0.924 + 1.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.98 + 8.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.04 - 5.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.814 - 1.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (4.60 - 7.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.55 - 2.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.19T + 83T^{2} \)
89 \( 1 + (-5.90 - 10.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.21T + 97T^{2} \)
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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

−9.416251125536624991310781665464, −8.544355813487853852436901832044, −8.019889720888730475654278142648, −6.98274163007728787758352480936, −6.29531143840758415387436160972, −5.48634257751092854215138128151, −4.21547836448463679222022634320, −2.77896969223186263240362503888, −1.87369625632118614688571595851, −0.64226233258619270938285292397, 2.52200936388216359750991216008, 3.34956821602600283514152822234, 3.78378376498182398456180682425, 4.72094951229926812756241824183, 6.35211215464424204235858565606, 7.29127478647729316713503305812, 7.65560419676664541180676781412, 8.907958024666530010754285081244, 9.556012131947323124823771558403, 10.40327378684257168930696787213

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