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Properties

Label 2-1001-1001.571-c0-0-9
Degree $2$
Conductor $1001$
Sign $-0.985 - 0.167i$
Analytic cond. $0.499564$
Root an. cond. $0.706798$
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 − 1.15i)2-s + (0.978 − 1.69i)3-s + (−0.395 + 0.684i)4-s − 2.61·6-s + (0.913 − 0.406i)7-s − 0.279·8-s + (−1.41 − 2.44i)9-s + (−0.5 + 0.866i)11-s + (0.773 + 1.34i)12-s + 13-s + (−1.08 − 0.786i)14-s + (0.582 + 1.00i)16-s + (−1.89 + 3.27i)18-s + (−0.309 − 0.535i)19-s + (0.204 − 1.94i)21-s + 1.33·22-s + ⋯
L(s)  = 1  + (−0.669 − 1.15i)2-s + (0.978 − 1.69i)3-s + (−0.395 + 0.684i)4-s − 2.61·6-s + (0.913 − 0.406i)7-s − 0.279·8-s + (−1.41 − 2.44i)9-s + (−0.5 + 0.866i)11-s + (0.773 + 1.34i)12-s + 13-s + (−1.08 − 0.786i)14-s + (0.582 + 1.00i)16-s + (−1.89 + 3.27i)18-s + (−0.309 − 0.535i)19-s + (0.204 − 1.94i)21-s + 1.33·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.985 - 0.167i$
Analytic conductor: \(0.499564\)
Root analytic conductor: \(0.706798\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :0),\ -0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023471244\)
\(L(\frac12)\) \(\approx\) \(1.023471244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.669 + 1.15i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 0.209T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.82T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.522117272626320853057058435359, −8.939747083663714567984986774601, −8.133799164494837654565315573496, −7.53309272830723762668268182571, −6.72898556195103827618720215973, −5.54250231386349912613705253295, −3.81414431267593564045869585381, −2.84456870432915912409703444442, −1.83301125196077417424929001087, −1.24108238321770155992389286182, 2.51360961361325412695400672897, 3.53481488399849844749850865736, 4.62941653134942687552717534482, 5.45578159777470808857796637133, 6.28624942780052549024350540905, 7.84200598708165240973565980618, 8.317269528238627306996087573382, 8.709758559736454440974410229329, 9.407229122645114795581990121432, 10.48915006093209395968395597696

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