Properties

Label 2-4001-1.1-c1-0-0
Degree $2$
Conductor $4001$
Sign $1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
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  − 2.80·2-s − 3.24·3-s + 5.84·4-s − 0.362·5-s + 9.08·6-s + 0.683·7-s − 10.7·8-s + 7.51·9-s + 1.01·10-s − 2.13·11-s − 18.9·12-s − 1.68·13-s − 1.91·14-s + 1.17·15-s + 18.4·16-s − 5.09·17-s − 21.0·18-s − 0.801·19-s − 2.11·20-s − 2.21·21-s + 5.97·22-s − 8.57·23-s + 34.9·24-s − 4.86·25-s + 4.72·26-s − 14.6·27-s + 3.99·28-s + ⋯
L(s)  = 1  − 1.98·2-s − 1.87·3-s + 2.92·4-s − 0.161·5-s + 3.70·6-s + 0.258·7-s − 3.80·8-s + 2.50·9-s + 0.320·10-s − 0.642·11-s − 5.47·12-s − 0.467·13-s − 0.511·14-s + 0.303·15-s + 4.62·16-s − 1.23·17-s − 4.96·18-s − 0.183·19-s − 0.473·20-s − 0.483·21-s + 1.27·22-s − 1.78·23-s + 7.13·24-s − 0.973·25-s + 0.925·26-s − 2.81·27-s + 0.754·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0003369582061\)
\(L(\frac12)\) \(\approx\) \(0.0003369582061\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 + 2.80T + 2T^{2} \)
3 \( 1 + 3.24T + 3T^{2} \)
5 \( 1 + 0.362T + 5T^{2} \)
7 \( 1 - 0.683T + 7T^{2} \)
11 \( 1 + 2.13T + 11T^{2} \)
13 \( 1 + 1.68T + 13T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 + 0.801T + 19T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
29 \( 1 + 3.50T + 29T^{2} \)
31 \( 1 - 1.80T + 31T^{2} \)
37 \( 1 + 5.00T + 37T^{2} \)
41 \( 1 + 9.31T + 41T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 - 7.78T + 53T^{2} \)
59 \( 1 + 1.98T + 59T^{2} \)
61 \( 1 + 3.99T + 61T^{2} \)
67 \( 1 - 5.71T + 67T^{2} \)
71 \( 1 - 8.29T + 71T^{2} \)
73 \( 1 + 1.66T + 73T^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + 1.86T + 83T^{2} \)
89 \( 1 + 6.35T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−8.304098637508850692431864306131, −7.81450108577376570839460464391, −6.95715352519918415668640991397, −6.54022205900463081228475607119, −5.78703455372106833400207771551, −5.07091438320978455950713194720, −3.84975293036613543694670327293, −2.22962248604289780971814110162, −1.56846978684756033099330802537, −0.01540561967072693204999701714, 0.01540561967072693204999701714, 1.56846978684756033099330802537, 2.22962248604289780971814110162, 3.84975293036613543694670327293, 5.07091438320978455950713194720, 5.78703455372106833400207771551, 6.54022205900463081228475607119, 6.95715352519918415668640991397, 7.81450108577376570839460464391, 8.304098637508850692431864306131

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