Properties

Label 2-4004-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.98·3-s + 2.28·5-s + 7-s + 5.89·9-s + 11-s − 13-s − 6.82·15-s + 7.65·17-s + 4.26·19-s − 2.98·21-s + 6.60·23-s + 0.237·25-s − 8.61·27-s + 4.02·29-s − 5.67·31-s − 2.98·33-s + 2.28·35-s + 2.12·37-s + 2.98·39-s + 5.12·41-s − 11.8·43-s + 13.4·45-s + 2.94·47-s + 49-s − 22.8·51-s + 14.4·53-s + 2.28·55-s + ⋯
L(s)  = 1  − 1.72·3-s + 1.02·5-s + 0.377·7-s + 1.96·9-s + 0.301·11-s − 0.277·13-s − 1.76·15-s + 1.85·17-s + 0.978·19-s − 0.650·21-s + 1.37·23-s + 0.0474·25-s − 1.65·27-s + 0.747·29-s − 1.01·31-s − 0.519·33-s + 0.386·35-s + 0.348·37-s + 0.477·39-s + 0.800·41-s − 1.80·43-s + 2.00·45-s + 0.429·47-s + 0.142·49-s − 3.19·51-s + 1.98·53-s + 0.308·55-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: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.603447705\)
\(L(\frac12)\) \(\approx\) \(1.603447705\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 - 4.26T + 19T^{2} \)
23 \( 1 - 6.60T + 23T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 + 5.67T + 31T^{2} \)
37 \( 1 - 2.12T + 37T^{2} \)
41 \( 1 - 5.12T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 2.94T + 47T^{2} \)
53 \( 1 - 14.4T + 53T^{2} \)
59 \( 1 - 9.55T + 59T^{2} \)
61 \( 1 - 8.38T + 61T^{2} \)
67 \( 1 + 0.770T + 67T^{2} \)
71 \( 1 + 8.20T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 7.34T + 79T^{2} \)
83 \( 1 - 0.774T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + 9.73T + 97T^{2} \)
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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

−8.465457425481413327555067176062, −7.29897278859038572709500252745, −6.99435491641155693436537340136, −5.94410516328892843384404479017, −5.46200481548827574576104944051, −5.15686703383587233786958321385, −4.09439264216335295555014241726, −2.91402410924748651064377223770, −1.52639234232923929506575501988, −0.893294190739507390801075173165, 0.893294190739507390801075173165, 1.52639234232923929506575501988, 2.91402410924748651064377223770, 4.09439264216335295555014241726, 5.15686703383587233786958321385, 5.46200481548827574576104944051, 5.94410516328892843384404479017, 6.99435491641155693436537340136, 7.29897278859038572709500252745, 8.465457425481413327555067176062

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