Properties

Label 2-4004-1.1-c1-0-42
Degree $2$
Conductor $4004$
Sign $-1$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.488·3-s − 3.65·5-s + 7-s − 2.76·9-s + 11-s + 13-s − 1.78·15-s + 1.65·17-s − 0.616·19-s + 0.488·21-s + 4.52·23-s + 8.37·25-s − 2.81·27-s + 3.33·29-s + 3.54·31-s + 0.488·33-s − 3.65·35-s − 9.82·37-s + 0.488·39-s + 1.93·41-s + 0.00506·43-s + 10.0·45-s + 4.15·47-s + 49-s + 0.808·51-s + 3.08·53-s − 3.65·55-s + ⋯
L(s)  = 1  + 0.281·3-s − 1.63·5-s + 0.377·7-s − 0.920·9-s + 0.301·11-s + 0.277·13-s − 0.460·15-s + 0.401·17-s − 0.141·19-s + 0.106·21-s + 0.943·23-s + 1.67·25-s − 0.541·27-s + 0.618·29-s + 0.637·31-s + 0.0849·33-s − 0.618·35-s − 1.61·37-s + 0.0781·39-s + 0.302·41-s + 0.000772·43-s + 1.50·45-s + 0.605·47-s + 0.142·49-s + 0.113·51-s + 0.423·53-s − 0.493·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 - 0.488T + 3T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 + 0.616T + 19T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 - 3.33T + 29T^{2} \)
31 \( 1 - 3.54T + 31T^{2} \)
37 \( 1 + 9.82T + 37T^{2} \)
41 \( 1 - 1.93T + 41T^{2} \)
43 \( 1 - 0.00506T + 43T^{2} \)
47 \( 1 - 4.15T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 9.09T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 7.88T + 67T^{2} \)
71 \( 1 + 7.09T + 71T^{2} \)
73 \( 1 - 1.62T + 73T^{2} \)
79 \( 1 + 2.27T + 79T^{2} \)
83 \( 1 + 2.98T + 83T^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
97 \( 1 + 0.425T + 97T^{2} \)
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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

−8.112670851926190485735173727452, −7.50796568544953921956542030756, −6.80562267735997535951527864877, −5.84050222686778956114995703351, −4.91242241679693865770921964797, −4.20422099648493053318122373227, −3.37949161426529445494788376707, −2.78204687161298935847381370150, −1.26771153294288235159964312295, 0, 1.26771153294288235159964312295, 2.78204687161298935847381370150, 3.37949161426529445494788376707, 4.20422099648493053318122373227, 4.91242241679693865770921964797, 5.84050222686778956114995703351, 6.80562267735997535951527864877, 7.50796568544953921956542030756, 8.112670851926190485735173727452

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