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Properties

Label 2-1001-7.2-c1-0-41
Degree $2$
Conductor $1001$
Sign $0.872 - 0.488i$
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  + (−0.738 + 1.27i)2-s + (0.252 + 0.438i)3-s + (−0.0920 − 0.159i)4-s + (0.747 − 1.29i)5-s − 0.747·6-s + (−2.63 − 0.203i)7-s − 2.68·8-s + (1.37 − 2.37i)9-s + (1.10 + 1.91i)10-s + (−0.5 − 0.866i)11-s + (0.0465 − 0.0806i)12-s − 13-s + (2.20 − 3.22i)14-s + 0.756·15-s + (2.16 − 3.75i)16-s + (1.60 + 2.77i)17-s + ⋯
L(s)  = 1  + (−0.522 + 0.905i)2-s + (0.146 + 0.252i)3-s + (−0.0460 − 0.0797i)4-s + (0.334 − 0.579i)5-s − 0.305·6-s + (−0.997 − 0.0769i)7-s − 0.948·8-s + (0.457 − 0.792i)9-s + (0.349 + 0.605i)10-s + (−0.150 − 0.261i)11-s + (0.0134 − 0.0232i)12-s − 0.277·13-s + (0.590 − 0.862i)14-s + 0.195·15-s + (0.541 − 0.938i)16-s + (0.388 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.872 - 0.488i$
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: \((2,\ 1001,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183920419\)
\(L(\frac12)\) \(\approx\) \(1.183920419\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.203i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.738 - 1.27i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.252 - 0.438i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.747 + 1.29i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (-1.60 - 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.48 + 4.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.454 - 0.787i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.05T + 29T^{2} \)
31 \( 1 + (-2.38 - 4.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.71 + 6.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.58T + 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 + (-6.44 + 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.10 + 7.10i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.29 + 3.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.481 - 0.834i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 + (6.95 + 12.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.64 + 2.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (0.282 - 0.488i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.08T + 97T^{2} \)
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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

−9.698635771268768771437250842362, −9.115509915436696469905506374553, −8.530479254054483079091140510806, −7.44204849455227028725429503571, −6.71270715634905381757582576222, −6.04168782975578269478897066336, −5.02401042245298509441152077497, −3.69753806084325696330275880957, −2.82160649154245068936369072103, −0.75451480767052401729229016446, 1.14559559131763479363441204841, 2.57427960525785862034023782150, 2.91441325039530877443952969859, 4.45829225621007225217878878122, 5.80850609605921179005565298698, 6.51099652553894830940582496674, 7.45092120511514735257444580821, 8.360615416921646867030894202797, 9.558481660085628057752759831316, 9.959693933083221832956190489890

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