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Properties

Label 2-1001-7.4-c1-0-0
Degree $2$
Conductor $1001$
Sign $0.0515 - 0.998i$
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  + (−1.38 − 2.40i)2-s + (−0.563 + 0.976i)3-s + (−2.84 + 4.93i)4-s + (−1.24 − 2.16i)5-s + 3.12·6-s + (1.34 + 2.27i)7-s + 10.2·8-s + (0.864 + 1.49i)9-s + (−3.46 + 5.99i)10-s + (−0.5 + 0.866i)11-s + (−3.20 − 5.55i)12-s − 13-s + (3.59 − 6.39i)14-s + 2.81·15-s + (−8.51 − 14.7i)16-s + (−0.157 + 0.273i)17-s + ⋯
L(s)  = 1  + (−0.980 − 1.69i)2-s + (−0.325 + 0.563i)3-s + (−1.42 + 2.46i)4-s + (−0.558 − 0.967i)5-s + 1.27·6-s + (0.510 + 0.860i)7-s + 3.62·8-s + (0.288 + 0.499i)9-s + (−1.09 + 1.89i)10-s + (−0.150 + 0.261i)11-s + (−0.926 − 1.60i)12-s − 0.277·13-s + (0.960 − 1.71i)14-s + 0.726·15-s + (−2.12 − 3.68i)16-s + (−0.0382 + 0.0662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.0515 - 0.998i$
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: \((2,\ 1001,\ (\ :1/2),\ 0.0515 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1673639639\)
\(L(\frac12)\) \(\approx\) \(0.1673639639\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.34 - 2.27i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.38 + 2.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.563 - 0.976i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.24 + 2.16i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (0.157 - 0.273i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.691 - 1.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.61 + 2.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.210T + 29T^{2} \)
31 \( 1 + (1.10 - 1.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.47 + 7.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.72T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-6.64 - 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.65 + 8.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.168 - 0.292i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.448 + 0.776i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.99 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.09T + 71T^{2} \)
73 \( 1 + (0.492 - 0.853i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.65 + 8.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.45T + 83T^{2} \)
89 \( 1 + (-5.54 - 9.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−10.23931932761862756969914179305, −9.490970664067397126205589636167, −8.701435725649184067322988392472, −8.207509712377165920050099509675, −7.38380969994390334511525588687, −5.28978652786735373430469728507, −4.62241646308092144176305154621, −3.86811916858436983470842970795, −2.49808199692677257494784623990, −1.50553364496884264254600225058, 0.12307450868415348698410964189, 1.45252504892145604959041021827, 3.73514748853210403557674063011, 4.89259934923580554735448620728, 5.88855519989236976008946975829, 6.93814138563617211553742626112, 7.04003736736988717748126230337, 7.82268677298627011740236821936, 8.582741715200418761152331223283, 9.692833741672587877198839723091

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