Properties

Label 2-2001-1.1-c1-0-40
Degree $2$
Conductor $2001$
Sign $1$
Analytic cond. $15.9780$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.161·2-s − 3-s − 1.97·4-s + 2.08·5-s − 0.161·6-s + 3.66·7-s − 0.641·8-s + 9-s + 0.335·10-s + 3.77·11-s + 1.97·12-s + 3.51·13-s + 0.592·14-s − 2.08·15-s + 3.84·16-s + 3.02·17-s + 0.161·18-s − 1.97·19-s − 4.10·20-s − 3.66·21-s + 0.609·22-s − 23-s + 0.641·24-s − 0.669·25-s + 0.566·26-s − 27-s − 7.24·28-s + ⋯
L(s)  = 1  + 0.114·2-s − 0.577·3-s − 0.986·4-s + 0.930·5-s − 0.0658·6-s + 1.38·7-s − 0.226·8-s + 0.333·9-s + 0.106·10-s + 1.13·11-s + 0.569·12-s + 0.973·13-s + 0.158·14-s − 0.537·15-s + 0.961·16-s + 0.733·17-s + 0.0380·18-s − 0.453·19-s − 0.918·20-s − 0.800·21-s + 0.130·22-s − 0.208·23-s + 0.130·24-s − 0.133·25-s + 0.111·26-s − 0.192·27-s − 1.36·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(15.9780\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.955140756\)
\(L(\frac12)\) \(\approx\) \(1.955140756\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 - 0.161T + 2T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 - 3.66T + 7T^{2} \)
11 \( 1 - 3.77T + 11T^{2} \)
13 \( 1 - 3.51T + 13T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 - 3.68T + 43T^{2} \)
47 \( 1 + 0.138T + 47T^{2} \)
53 \( 1 - 3.01T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 + 3.28T + 71T^{2} \)
73 \( 1 - 1.53T + 73T^{2} \)
79 \( 1 - 7.31T + 79T^{2} \)
83 \( 1 - 5.82T + 83T^{2} \)
89 \( 1 - 6.06T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−9.133576348685573866291757514431, −8.494603394019725810842792595726, −7.74991396206926494642127033584, −6.53849019069357684067823765589, −5.81413852474450680801122606906, −5.20490180986062787144507633454, −4.36546291626672335969150301496, −3.60997181330544746110685169345, −1.83958028056890092677438658620, −1.07371161253457200134156737816, 1.07371161253457200134156737816, 1.83958028056890092677438658620, 3.60997181330544746110685169345, 4.36546291626672335969150301496, 5.20490180986062787144507633454, 5.81413852474450680801122606906, 6.53849019069357684067823765589, 7.74991396206926494642127033584, 8.494603394019725810842792595726, 9.133576348685573866291757514431

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