Properties

Label 2-2001-1.1-c1-0-47
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 1.58·5-s − 3.08·7-s + 9-s + 6.03·11-s + 2·12-s − 2.36·13-s + 1.58·15-s + 4·16-s + 6.85·17-s − 5.13·19-s + 3.16·20-s + 3.08·21-s + 23-s − 2.50·25-s − 27-s + 6.16·28-s + 29-s + 0.0784·31-s − 6.03·33-s + 4.87·35-s − 2·36-s + 5.42·37-s + 2.36·39-s + 7.95·41-s − 2.05·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.706·5-s − 1.16·7-s + 0.333·9-s + 1.81·11-s + 0.577·12-s − 0.657·13-s + 0.408·15-s + 16-s + 1.66·17-s − 1.17·19-s + 0.706·20-s + 0.672·21-s + 0.208·23-s − 0.500·25-s − 0.192·27-s + 1.16·28-s + 0.185·29-s + 0.0140·31-s − 1.05·33-s + 0.823·35-s − 0.333·36-s + 0.892·37-s + 0.379·39-s + 1.24·41-s − 0.313·43-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: \(1\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 - 6.03T + 11T^{2} \)
13 \( 1 + 2.36T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
31 \( 1 - 0.0784T + 31T^{2} \)
37 \( 1 - 5.42T + 37T^{2} \)
41 \( 1 - 7.95T + 41T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + 5.00T + 53T^{2} \)
59 \( 1 + 0.756T + 59T^{2} \)
61 \( 1 + 3.79T + 61T^{2} \)
67 \( 1 - 6.16T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 2.91T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 5.48T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 2.53T + 97T^{2} \)
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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

−8.918325944322273856967751359152, −7.992981097090725045461239846005, −7.17001901798670917947504647815, −6.28611956699328826844239477422, −5.68500440782437704851733296548, −4.43296074746332024618418378238, −3.98218699978229777054595998930, −3.12061721193960262082869744068, −1.18040099070440492574044301542, 0, 1.18040099070440492574044301542, 3.12061721193960262082869744068, 3.98218699978229777054595998930, 4.43296074746332024618418378238, 5.68500440782437704851733296548, 6.28611956699328826844239477422, 7.17001901798670917947504647815, 7.992981097090725045461239846005, 8.918325944322273856967751359152

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