Properties

Label 2-4004-1.1-c1-0-25
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  − 0.554·3-s + 3.11·5-s + 7-s − 2.69·9-s − 11-s + 13-s − 1.72·15-s − 0.195·17-s + 2.90·19-s − 0.554·21-s + 2.28·23-s + 4.69·25-s + 3.15·27-s − 2.31·29-s − 4.38·31-s + 0.554·33-s + 3.11·35-s + 1.83·37-s − 0.554·39-s + 12.6·41-s + 0.100·43-s − 8.38·45-s − 11.4·47-s + 49-s + 0.108·51-s + 9.79·53-s − 3.11·55-s + ⋯
L(s)  = 1  − 0.320·3-s + 1.39·5-s + 0.377·7-s − 0.897·9-s − 0.301·11-s + 0.277·13-s − 0.446·15-s − 0.0473·17-s + 0.667·19-s − 0.121·21-s + 0.476·23-s + 0.939·25-s + 0.607·27-s − 0.430·29-s − 0.787·31-s + 0.0965·33-s + 0.526·35-s + 0.301·37-s − 0.0888·39-s + 1.97·41-s + 0.0153·43-s − 1.24·45-s − 1.66·47-s + 0.142·49-s + 0.0151·51-s + 1.34·53-s − 0.419·55-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

\(L(1)\) \(\approx\) \(2.216796402\)
\(L(\frac12)\) \(\approx\) \(2.216796402\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.554T + 3T^{2} \)
5 \( 1 - 3.11T + 5T^{2} \)
17 \( 1 + 0.195T + 17T^{2} \)
19 \( 1 - 2.90T + 19T^{2} \)
23 \( 1 - 2.28T + 23T^{2} \)
29 \( 1 + 2.31T + 29T^{2} \)
31 \( 1 + 4.38T + 31T^{2} \)
37 \( 1 - 1.83T + 37T^{2} \)
41 \( 1 - 12.6T + 41T^{2} \)
43 \( 1 - 0.100T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 9.79T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 3.29T + 61T^{2} \)
67 \( 1 - 9.20T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 8.50T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 0.0580T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.518836390401322016888222058809, −7.75924976060804155286431185415, −6.82539904265649564396244643709, −6.09087250773445040757188907251, −5.43976309047423920549547466982, −5.08880847334470078826666687397, −3.81397800930180219978553015021, −2.74576932846564588632036607666, −2.03536330982297771478332068973, −0.881845823439036896017926428357, 0.881845823439036896017926428357, 2.03536330982297771478332068973, 2.74576932846564588632036607666, 3.81397800930180219978553015021, 5.08880847334470078826666687397, 5.43976309047423920549547466982, 6.09087250773445040757188907251, 6.82539904265649564396244643709, 7.75924976060804155286431185415, 8.518836390401322016888222058809

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