Properties

Label 2-4001-1.1-c1-0-112
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  − 0.538·2-s − 1.14·3-s − 1.70·4-s + 0.955·5-s + 0.618·6-s + 4.48·7-s + 1.99·8-s − 1.68·9-s − 0.514·10-s + 0.959·11-s + 1.96·12-s − 2.43·13-s − 2.41·14-s − 1.09·15-s + 2.34·16-s + 3.03·17-s + 0.907·18-s + 7.55·19-s − 1.63·20-s − 5.14·21-s − 0.517·22-s + 0.535·23-s − 2.29·24-s − 4.08·25-s + 1.31·26-s + 5.37·27-s − 7.67·28-s + ⋯
L(s)  = 1  − 0.380·2-s − 0.662·3-s − 0.854·4-s + 0.427·5-s + 0.252·6-s + 1.69·7-s + 0.706·8-s − 0.561·9-s − 0.162·10-s + 0.289·11-s + 0.566·12-s − 0.675·13-s − 0.646·14-s − 0.282·15-s + 0.585·16-s + 0.736·17-s + 0.213·18-s + 1.73·19-s − 0.365·20-s − 1.12·21-s − 0.110·22-s + 0.111·23-s − 0.468·24-s − 0.817·25-s + 0.257·26-s + 1.03·27-s − 1.44·28-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

\(L(1)\) \(\approx\) \(1.332195257\)
\(L(\frac12)\) \(\approx\) \(1.332195257\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.538T + 2T^{2} \)
3 \( 1 + 1.14T + 3T^{2} \)
5 \( 1 - 0.955T + 5T^{2} \)
7 \( 1 - 4.48T + 7T^{2} \)
11 \( 1 - 0.959T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 - 3.03T + 17T^{2} \)
19 \( 1 - 7.55T + 19T^{2} \)
23 \( 1 - 0.535T + 23T^{2} \)
29 \( 1 - 5.42T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 - 9.04T + 41T^{2} \)
43 \( 1 - 5.01T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 - 6.74T + 59T^{2} \)
61 \( 1 + 2.62T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 + 6.66T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 6.98T + 83T^{2} \)
89 \( 1 - 9.51T + 89T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−8.396778167603412871342804061756, −7.78101176667819896526909638552, −7.29130214008625152983684053970, −5.89223852450139132317964919382, −5.42072732670841584402864128639, −4.88389170200984194841718163184, −4.12591028451049633340473252167, −2.87135785802283565083821969169, −1.59362200720779506999637268811, −0.798835754535184961063429769858, 0.798835754535184961063429769858, 1.59362200720779506999637268811, 2.87135785802283565083821969169, 4.12591028451049633340473252167, 4.88389170200984194841718163184, 5.42072732670841584402864128639, 5.89223852450139132317964919382, 7.29130214008625152983684053970, 7.78101176667819896526909638552, 8.396778167603412871342804061756

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