Properties

Label 2-4001-1.1-c1-0-11
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  − 2.73·2-s + 2.60·3-s + 5.46·4-s − 2.78·5-s − 7.11·6-s − 4.77·7-s − 9.45·8-s + 3.79·9-s + 7.59·10-s − 5.83·11-s + 14.2·12-s − 1.87·13-s + 13.0·14-s − 7.24·15-s + 14.8·16-s − 7.42·17-s − 10.3·18-s − 2.84·19-s − 15.1·20-s − 12.4·21-s + 15.9·22-s + 0.931·23-s − 24.6·24-s + 2.72·25-s + 5.11·26-s + 2.07·27-s − 26.0·28-s + ⋯
L(s)  = 1  − 1.93·2-s + 1.50·3-s + 2.73·4-s − 1.24·5-s − 2.90·6-s − 1.80·7-s − 3.34·8-s + 1.26·9-s + 2.40·10-s − 1.75·11-s + 4.10·12-s − 0.519·13-s + 3.48·14-s − 1.87·15-s + 3.72·16-s − 1.80·17-s − 2.44·18-s − 0.651·19-s − 3.39·20-s − 2.71·21-s + 3.39·22-s + 0.194·23-s − 5.02·24-s + 0.545·25-s + 1.00·26-s + 0.398·27-s − 4.92·28-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

\(L(1)\) \(\approx\) \(0.05714155746\)
\(L(\frac12)\) \(\approx\) \(0.05714155746\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 2T^{2} \)
3 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 + 5.83T + 11T^{2} \)
13 \( 1 + 1.87T + 13T^{2} \)
17 \( 1 + 7.42T + 17T^{2} \)
19 \( 1 + 2.84T + 19T^{2} \)
23 \( 1 - 0.931T + 23T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 - 9.52T + 31T^{2} \)
37 \( 1 + 9.70T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 - 4.05T + 43T^{2} \)
47 \( 1 - 1.91T + 47T^{2} \)
53 \( 1 - 0.163T + 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + 4.17T + 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 - 4.79T + 79T^{2} \)
83 \( 1 - 1.13T + 83T^{2} \)
89 \( 1 + 6.41T + 89T^{2} \)
97 \( 1 + 3.59T + 97T^{2} \)
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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

−8.539285684880577802249590033333, −7.897942728585938974758443290020, −7.33348176790229650403491429180, −6.90509432977258349412286557482, −5.95710660553339439823512809958, −4.28005263869965396618088580724, −3.23674933191276667138967591437, −2.74347116773422885560323378781, −2.14512616305068917704813082541, −0.15132079895554695238645358923, 0.15132079895554695238645358923, 2.14512616305068917704813082541, 2.74347116773422885560323378781, 3.23674933191276667138967591437, 4.28005263869965396618088580724, 5.95710660553339439823512809958, 6.90509432977258349412286557482, 7.33348176790229650403491429180, 7.897942728585938974758443290020, 8.539285684880577802249590033333

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