Properties

Label 2-4001-1.1-c1-0-1
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  − 0.114·2-s + 0.742·3-s − 1.98·4-s − 1.13·5-s − 0.0853·6-s − 2.76·7-s + 0.458·8-s − 2.44·9-s + 0.130·10-s − 5.26·11-s − 1.47·12-s − 5.84·13-s + 0.317·14-s − 0.841·15-s + 3.92·16-s − 0.422·17-s + 0.281·18-s − 0.824·19-s + 2.25·20-s − 2.04·21-s + 0.605·22-s + 3.44·23-s + 0.340·24-s − 3.71·25-s + 0.671·26-s − 4.04·27-s + 5.48·28-s + ⋯
L(s)  = 1  − 0.0813·2-s + 0.428·3-s − 0.993·4-s − 0.506·5-s − 0.0348·6-s − 1.04·7-s + 0.162·8-s − 0.816·9-s + 0.0412·10-s − 1.58·11-s − 0.425·12-s − 1.61·13-s + 0.0848·14-s − 0.217·15-s + 0.980·16-s − 0.102·17-s + 0.0663·18-s − 0.189·19-s + 0.503·20-s − 0.447·21-s + 0.129·22-s + 0.717·23-s + 0.0694·24-s − 0.742·25-s + 0.131·26-s − 0.778·27-s + 1.03·28-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

\(L(1)\) \(\approx\) \(0.0008945480070\)
\(L(\frac12)\) \(\approx\) \(0.0008945480070\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.114T + 2T^{2} \)
3 \( 1 - 0.742T + 3T^{2} \)
5 \( 1 + 1.13T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 + 5.26T + 11T^{2} \)
13 \( 1 + 5.84T + 13T^{2} \)
17 \( 1 + 0.422T + 17T^{2} \)
19 \( 1 + 0.824T + 19T^{2} \)
23 \( 1 - 3.44T + 23T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 - 2.65T + 31T^{2} \)
37 \( 1 + 6.68T + 37T^{2} \)
41 \( 1 + 3.98T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 3.54T + 47T^{2} \)
53 \( 1 + 5.28T + 53T^{2} \)
59 \( 1 + 0.983T + 59T^{2} \)
61 \( 1 - 1.72T + 61T^{2} \)
67 \( 1 - 2.77T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 5.78T + 73T^{2} \)
79 \( 1 - 0.287T + 79T^{2} \)
83 \( 1 - 2.08T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 1.82T + 97T^{2} \)
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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

−8.496247004743205087455002710072, −7.75499968942466468648199861878, −7.30641735549044522108677949865, −6.17668816045243159582701043218, −5.18376303959556518553405653475, −4.89372586374907577249243822460, −3.62502485894195735959023040756, −3.10225480066630781210394258874, −2.22102103875291069700402747212, −0.01561392479398394160029485851, 0.01561392479398394160029485851, 2.22102103875291069700402747212, 3.10225480066630781210394258874, 3.62502485894195735959023040756, 4.89372586374907577249243822460, 5.18376303959556518553405653475, 6.17668816045243159582701043218, 7.30641735549044522108677949865, 7.75499968942466468648199861878, 8.496247004743205087455002710072

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