Properties

Label 2-2001-1.1-c1-0-60
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  + 2.74·2-s − 3-s + 5.53·4-s + 0.0839·5-s − 2.74·6-s − 2.30·7-s + 9.69·8-s + 9-s + 0.230·10-s + 4.15·11-s − 5.53·12-s − 0.439·13-s − 6.31·14-s − 0.0839·15-s + 15.5·16-s − 0.388·17-s + 2.74·18-s + 5.29·19-s + 0.464·20-s + 2.30·21-s + 11.4·22-s − 23-s − 9.69·24-s − 4.99·25-s − 1.20·26-s − 27-s − 12.7·28-s + ⋯
L(s)  = 1  + 1.94·2-s − 0.577·3-s + 2.76·4-s + 0.0375·5-s − 1.12·6-s − 0.869·7-s + 3.42·8-s + 0.333·9-s + 0.0728·10-s + 1.25·11-s − 1.59·12-s − 0.121·13-s − 1.68·14-s − 0.0216·15-s + 3.88·16-s − 0.0941·17-s + 0.646·18-s + 1.21·19-s + 0.103·20-s + 0.502·21-s + 2.43·22-s − 0.208·23-s − 1.97·24-s − 0.998·25-s − 0.236·26-s − 0.192·27-s − 2.40·28-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

\(L(1)\) \(\approx\) \(5.325093639\)
\(L(\frac12)\) \(\approx\) \(5.325093639\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.74T + 2T^{2} \)
5 \( 1 - 0.0839T + 5T^{2} \)
7 \( 1 + 2.30T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 + 0.439T + 13T^{2} \)
17 \( 1 + 0.388T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 - 9.04T + 37T^{2} \)
41 \( 1 + 0.159T + 41T^{2} \)
43 \( 1 + 6.03T + 43T^{2} \)
47 \( 1 + 0.396T + 47T^{2} \)
53 \( 1 - 2.43T + 53T^{2} \)
59 \( 1 + 4.08T + 59T^{2} \)
61 \( 1 - 9.28T + 61T^{2} \)
67 \( 1 - 4.55T + 67T^{2} \)
71 \( 1 + 3.85T + 71T^{2} \)
73 \( 1 - 0.0113T + 73T^{2} \)
79 \( 1 - 9.57T + 79T^{2} \)
83 \( 1 + 5.55T + 83T^{2} \)
89 \( 1 + 4.02T + 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−9.522442316506097086713418912814, −7.975062123524895503294779984484, −7.03958259111711660977515600234, −6.50155505733815646258772121503, −5.90173306848395525609320976761, −5.15560033256851393634002527087, −4.18892830599786945452249837166, −3.61105935891614539283593226726, −2.65441970734118153102397809031, −1.36229247042027200949634847405, 1.36229247042027200949634847405, 2.65441970734118153102397809031, 3.61105935891614539283593226726, 4.18892830599786945452249837166, 5.15560033256851393634002527087, 5.90173306848395525609320976761, 6.50155505733815646258772121503, 7.03958259111711660977515600234, 7.975062123524895503294779984484, 9.522442316506097086713418912814

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