Properties

Label 2-4001-1.1-c1-0-135
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  + 0.799·2-s − 0.343·3-s − 1.36·4-s + 3.87·5-s − 0.274·6-s + 3.67·7-s − 2.68·8-s − 2.88·9-s + 3.09·10-s + 3.19·11-s + 0.467·12-s − 5.30·13-s + 2.93·14-s − 1.33·15-s + 0.575·16-s − 1.86·17-s − 2.30·18-s − 2.90·19-s − 5.27·20-s − 1.26·21-s + 2.55·22-s − 3.28·23-s + 0.923·24-s + 10.0·25-s − 4.23·26-s + 2.02·27-s − 4.99·28-s + ⋯
L(s)  = 1  + 0.565·2-s − 0.198·3-s − 0.680·4-s + 1.73·5-s − 0.112·6-s + 1.38·7-s − 0.949·8-s − 0.960·9-s + 0.979·10-s + 0.962·11-s + 0.135·12-s − 1.47·13-s + 0.783·14-s − 0.343·15-s + 0.143·16-s − 0.453·17-s − 0.542·18-s − 0.666·19-s − 1.17·20-s − 0.275·21-s + 0.543·22-s − 0.684·23-s + 0.188·24-s + 2.00·25-s − 0.831·26-s + 0.389·27-s − 0.944·28-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

\(L(1)\) \(\approx\) \(2.891358865\)
\(L(\frac12)\) \(\approx\) \(2.891358865\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.799T + 2T^{2} \)
3 \( 1 + 0.343T + 3T^{2} \)
5 \( 1 - 3.87T + 5T^{2} \)
7 \( 1 - 3.67T + 7T^{2} \)
11 \( 1 - 3.19T + 11T^{2} \)
13 \( 1 + 5.30T + 13T^{2} \)
17 \( 1 + 1.86T + 17T^{2} \)
19 \( 1 + 2.90T + 19T^{2} \)
23 \( 1 + 3.28T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 5.18T + 41T^{2} \)
43 \( 1 - 9.49T + 43T^{2} \)
47 \( 1 - 8.49T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 6.65T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 + 3.20T + 67T^{2} \)
71 \( 1 + 5.41T + 71T^{2} \)
73 \( 1 - 5.64T + 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 - 0.854T + 83T^{2} \)
89 \( 1 + 8.87T + 89T^{2} \)
97 \( 1 - 6.76T + 97T^{2} \)
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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

−8.780073311396209564318381208603, −7.80505573000856290891969715068, −6.65360441435251370153750653804, −5.97234235687817244356235595963, −5.47690737362185499590690517674, −4.70521529424060465381552617697, −4.27445532800415361448489526314, −2.68335254251592676766043791362, −2.21002558953799022561149195323, −0.933658214600414393494774710756, 0.933658214600414393494774710756, 2.21002558953799022561149195323, 2.68335254251592676766043791362, 4.27445532800415361448489526314, 4.70521529424060465381552617697, 5.47690737362185499590690517674, 5.97234235687817244356235595963, 6.65360441435251370153750653804, 7.80505573000856290891969715068, 8.780073311396209564318381208603

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