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Properties

Label 2-1001-7.4-c1-0-1
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  + (−0.834 − 1.44i)2-s + (−1.40 + 2.43i)3-s + (−0.393 + 0.681i)4-s + (1.40 + 2.43i)5-s + 4.69·6-s + (−2 + 1.73i)7-s − 2.02·8-s + (−2.45 − 4.25i)9-s + (2.34 − 4.06i)10-s + (0.5 − 0.866i)11-s + (−1.10 − 1.91i)12-s − 13-s + (4.17 + 1.44i)14-s − 7.90·15-s + (2.47 + 4.29i)16-s + (−2.51 + 4.35i)17-s + ⋯
L(s)  = 1  + (−0.590 − 1.02i)2-s + (−0.811 + 1.40i)3-s + (−0.196 + 0.340i)4-s + (0.628 + 1.08i)5-s + 1.91·6-s + (−0.755 + 0.654i)7-s − 0.715·8-s + (−0.817 − 1.41i)9-s + (0.742 − 1.28i)10-s + (0.150 − 0.261i)11-s + (−0.319 − 0.553i)12-s − 0.277·13-s + (1.11 + 0.386i)14-s − 2.04·15-s + (0.619 + 1.07i)16-s + (−0.609 + 1.05i)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

\(L(1)\) \(\approx\) \(0.2639519970\)
\(L(\frac12)\) \(\approx\) \(0.2639519970\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.834 + 1.44i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.40 - 2.43i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.40 - 2.43i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (2.51 - 4.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.271 + 0.470i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.974T + 29T^{2} \)
31 \( 1 + (-1.02 + 1.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.02 + 1.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.355T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + (0.441 + 0.764i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.53 + 7.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.83 + 8.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.13 + 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.14 - 7.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + (-5.56 + 9.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.97 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (-0.327 - 0.567i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.952T + 97T^{2} \)
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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

−10.25668825738115936049466268328, −9.950326735028331503728078929652, −9.321788337990269209445981966197, −8.403942073681504982131578243280, −6.61528950561270549485849100728, −6.06570735493139220036559800753, −5.35654303322204670661153080213, −3.82042591414134241989307281571, −3.15795965127436589787455842325, −2.03272947816900124217668441931, 0.17257459404037613051318386196, 1.22471206379033969766324618253, 2.77972782441176388148755721982, 4.75099745917082462340362015345, 5.58961738338954304210590609001, 6.38474340615680388320300146420, 7.14491938056734749548271163502, 7.40755895295894992751759815507, 8.643505179169878911333119464334, 9.247938663845343353645552994475

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