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Properties

Label 2-1001-7.2-c1-0-72
Degree $2$
Conductor $1001$
Sign $-0.989 + 0.146i$
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  + (1.11 − 1.93i)2-s + (−0.407 − 0.705i)3-s + (−1.49 − 2.59i)4-s + (1.32 − 2.30i)5-s − 1.82·6-s + (2.39 + 1.12i)7-s − 2.22·8-s + (1.16 − 2.02i)9-s + (−2.97 − 5.14i)10-s + (−0.5 − 0.866i)11-s + (−1.22 + 2.11i)12-s − 13-s + (4.85 − 3.37i)14-s − 2.16·15-s + (0.510 − 0.883i)16-s + (2.78 + 4.81i)17-s + ⋯
L(s)  = 1  + (0.790 − 1.36i)2-s + (−0.235 − 0.407i)3-s + (−0.748 − 1.29i)4-s + (0.594 − 1.02i)5-s − 0.743·6-s + (0.904 + 0.425i)7-s − 0.786·8-s + (0.389 − 0.674i)9-s + (−0.939 − 1.62i)10-s + (−0.150 − 0.261i)11-s + (−0.352 + 0.610i)12-s − 0.277·13-s + (1.29 − 0.902i)14-s − 0.559·15-s + (0.127 − 0.220i)16-s + (0.674 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.989 + 0.146i$
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: \((2,\ 1001,\ (\ :1/2),\ -0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.780033309\)
\(L(\frac12)\) \(\approx\) \(2.780033309\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.39 - 1.12i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-1.11 + 1.93i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.407 + 0.705i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.32 + 2.30i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (-2.78 - 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.46 + 2.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.697 - 1.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.282T + 29T^{2} \)
31 \( 1 + (-4.07 - 7.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.40 - 5.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (-2.70 + 4.68i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.359 + 0.622i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.76 - 8.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.91 - 8.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.20 - 7.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.0651T + 71T^{2} \)
73 \( 1 + (1.43 + 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.760 - 1.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 + (7.83 - 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.62T + 97T^{2} \)
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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

−9.895430925682159635684556762226, −8.897524255858537990538885686563, −8.179148103916933793253328508840, −6.86690207539644720944596043073, −5.58766886190281897622043498481, −5.15396945773267325232749501398, −4.21159613635573131811755694864, −3.08435220732430438368652102646, −1.70211192978794054538990316697, −1.20131889533475472802163698802, 2.05304783219129221901140813186, 3.51803302343601244848527537442, 4.66689107095711845634560894741, 5.15728334903568600196891574956, 6.04519035934231294793077295111, 7.01140931314247401818729186941, 7.55059271782232439179727913935, 8.216492067304818740398017266860, 9.727176886829109166336414501479, 10.27914931205425251102057020241

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