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Properties

Label 2-1001-7.2-c1-0-14
Degree $2$
Conductor $1001$
Sign $0.754 - 0.656i$
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  + (−1.06 + 1.85i)2-s + (−1.15 − 1.99i)3-s + (−1.28 − 2.23i)4-s + (−1.42 + 2.47i)5-s + 4.92·6-s + (−2.62 + 0.341i)7-s + 1.23·8-s + (−1.15 + 1.99i)9-s + (−3.05 − 5.28i)10-s + (−0.5 − 0.866i)11-s + (−2.96 + 5.14i)12-s − 13-s + (2.17 − 5.22i)14-s + 6.57·15-s + (1.25 − 2.17i)16-s + (−1.86 − 3.22i)17-s + ⋯
L(s)  = 1  + (−0.756 + 1.31i)2-s + (−0.664 − 1.15i)3-s + (−0.644 − 1.11i)4-s + (−0.638 + 1.10i)5-s + 2.01·6-s + (−0.991 + 0.128i)7-s + 0.436·8-s + (−0.384 + 0.665i)9-s + (−0.965 − 1.67i)10-s + (−0.150 − 0.261i)11-s + (−0.856 + 1.48i)12-s − 0.277·13-s + (0.581 − 1.39i)14-s + 1.69·15-s + (0.314 − 0.543i)16-s + (−0.452 − 0.783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.754 - 0.656i$
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: \((2,\ 1001,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3424591082\)
\(L(\frac12)\) \(\approx\) \(0.3424591082\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 - 0.341i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.06 - 1.85i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.15 + 1.99i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.42 - 2.47i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (1.86 + 3.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0757 - 0.131i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.18 - 5.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + (1.75 + 3.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.59 + 4.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (4.59 - 7.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.72 + 2.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.64 - 9.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.04 + 5.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.39 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 + (4.41 + 7.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.296 - 0.512i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.04T + 83T^{2} \)
89 \( 1 + (-6.90 + 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−9.673830320149278062963386516031, −9.234222169537877266657586061353, −7.81447318555436012102381686142, −7.37477899616579812205438276582, −6.98864094302194218194995994636, −6.00613143865470600592118286868, −5.70744307118888673435819484806, −3.77987802371604721121100169185, −2.51647888857054307696173367087, −0.44427340208577712742951637856, 0.56291040761376684919125894977, 2.28338688657558111027661903038, 3.79155496050017925149791902516, 4.14389788723114554863710122408, 5.26627022488692792773431046548, 6.30326640650605830255544501054, 7.78033228403723704979869513635, 8.689149337624293730512517761475, 9.439974738479483485377318012435, 9.834950644710029500644864106502

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