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Properties

Label 2-1001-7.4-c1-0-58
Degree $2$
Conductor $1001$
Sign $-0.197 + 0.980i$
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  + (−0.0543 − 0.0940i)2-s + (−1.02 + 1.77i)3-s + (0.994 − 1.72i)4-s + (0.123 + 0.213i)5-s + 0.222·6-s + (−1.00 − 2.44i)7-s − 0.433·8-s + (−0.602 − 1.04i)9-s + (0.0133 − 0.0231i)10-s + (−0.5 + 0.866i)11-s + (2.03 + 3.53i)12-s − 13-s + (−0.176 + 0.227i)14-s − 0.505·15-s + (−1.96 − 3.40i)16-s + (0.255 − 0.441i)17-s + ⋯
L(s)  = 1  + (−0.0384 − 0.0665i)2-s + (−0.591 + 1.02i)3-s + (0.497 − 0.860i)4-s + (0.0551 + 0.0954i)5-s + 0.0909·6-s + (−0.378 − 0.925i)7-s − 0.153·8-s + (−0.200 − 0.347i)9-s + (0.00423 − 0.00733i)10-s + (−0.150 + 0.261i)11-s + (0.588 + 1.01i)12-s − 0.277·13-s + (−0.0470 + 0.0607i)14-s − 0.130·15-s + (−0.491 − 0.850i)16-s + (0.0618 − 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.197 + 0.980i$
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: \((2,\ 1001,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8213762463\)
\(L(\frac12)\) \(\approx\) \(0.8213762463\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.00 + 2.44i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.0543 + 0.0940i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.02 - 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.123 - 0.213i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (-0.255 + 0.441i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.438 + 0.759i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.49 + 6.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.01T + 29T^{2} \)
31 \( 1 + (3.16 - 5.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.63 + 6.29i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
43 \( 1 + 5.79T + 43T^{2} \)
47 \( 1 + (-1.23 - 2.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.12 + 7.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.81 + 8.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.35 + 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.98 + 5.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.36T + 71T^{2} \)
73 \( 1 + (-4.73 + 8.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.87 - 4.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.58T + 83T^{2} \)
89 \( 1 + (-3.05 - 5.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−10.01958023533765739135162925904, −9.341162086933168901134880958249, −8.074842073092968062353300351453, −6.86491048845656800918880713671, −6.41050801910750262534742942779, −5.18065947253750485579165486155, −4.67039921914002394054154667365, −3.54362146372766847800059625650, −2.12655061212841348355980788529, −0.38969958787636389128333948199, 1.61952076654209024319174084582, 2.72293793973291228409531886867, 3.77671160237124172825824686461, 5.36610856299363430429663171566, 6.07313596684366342683278090015, 6.86905747911543214169082919746, 7.55982070047402485319600415415, 8.380850498419782661217600295048, 9.208444830108674100957623437865, 10.25297496901038400400399925262

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