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Properties

Label 2-1001-7.4-c1-0-37
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  + (−1.18 − 2.04i)2-s + (0.0817 − 0.141i)3-s + (−1.79 + 3.10i)4-s + (−0.0817 − 0.141i)5-s − 0.386·6-s + (−2 + 1.73i)7-s + 3.75·8-s + (1.48 + 2.57i)9-s + (−0.193 + 0.334i)10-s + (0.5 − 0.866i)11-s + (0.293 + 0.508i)12-s − 13-s + (5.90 + 2.04i)14-s − 0.0267·15-s + (−0.845 − 1.46i)16-s + (0.375 − 0.649i)17-s + ⋯
L(s)  = 1  + (−0.835 − 1.44i)2-s + (0.0472 − 0.0817i)3-s + (−0.896 + 1.55i)4-s + (−0.0365 − 0.0633i)5-s − 0.157·6-s + (−0.755 + 0.654i)7-s + 1.32·8-s + (0.495 + 0.858i)9-s + (−0.0611 + 0.105i)10-s + (0.150 − 0.261i)11-s + (0.0846 + 0.146i)12-s − 0.277·13-s + (1.57 + 0.547i)14-s − 0.00690·15-s + (−0.211 − 0.366i)16-s + (0.0909 − 0.157i)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

\(L(1)\) \(\approx\) \(0.5924213992\)
\(L(\frac12)\) \(\approx\) \(0.5924213992\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 2.04i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0817 + 0.141i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.0817 + 0.141i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (-0.375 + 0.649i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.43 + 5.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.47 - 2.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.75T + 29T^{2} \)
31 \( 1 + (-5.39 + 9.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.64 + 2.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.11T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 + (-0.611 - 1.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.01 + 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.91 - 6.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.39 + 7.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.91 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.817T + 71T^{2} \)
73 \( 1 + (1.43 - 2.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.65 - 4.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.02T + 83T^{2} \)
89 \( 1 + (7.80 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.282T + 97T^{2} \)
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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

−9.638722642016366250646260812442, −9.034714503815926649159620362546, −8.283369793947746567405090269177, −7.32955267029386002636828598374, −6.22530277840504590835867906876, −4.94150359276218320817071399618, −3.84679141306469630971877023886, −2.69965535637817066070675472200, −2.05912350808478815415480649097, −0.40288710303739366681501570728, 1.19865833515620129766655747683, 3.33239020998424179710683649760, 4.36661024569295568446513468270, 5.58186566605500679520822411072, 6.52708641940981915593479272486, 6.93450836775147381998727635541, 7.72650730371060281371041583851, 8.710711972461673878888224519514, 9.333016979467854374468465742489, 10.13421534810801055327493734345

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