Properties

Label 2-4004-77.76-c1-0-49
Degree $2$
Conductor $4004$
Sign $0.771 - 0.636i$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41i·3-s − 2.25i·5-s + (2.16 + 1.52i)7-s − 2.82·9-s + (0.870 − 3.20i)11-s + 13-s + 5.44·15-s + 1.61·17-s − 2.82·19-s + (−3.68 + 5.21i)21-s − 1.64·23-s − 0.0911·25-s + 0.429i·27-s + 0.879i·29-s + 0.464i·31-s + ⋯
L(s)  = 1  + 1.39i·3-s − 1.00i·5-s + (0.816 + 0.577i)7-s − 0.940·9-s + (0.262 − 0.964i)11-s + 0.277·13-s + 1.40·15-s + 0.390·17-s − 0.647·19-s + (−0.803 + 1.13i)21-s − 0.343·23-s − 0.0182·25-s + 0.0826i·27-s + 0.163i·29-s + 0.0833i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.771 - 0.636i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (3849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.204358689\)
\(L(\frac12)\) \(\approx\) \(2.204358689\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 1.52i)T \)
11 \( 1 + (-0.870 + 3.20i)T \)
13 \( 1 - T \)
good3 \( 1 - 2.41iT - 3T^{2} \)
5 \( 1 + 2.25iT - 5T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 - 0.879iT - 29T^{2} \)
31 \( 1 - 0.464iT - 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 - 6.38T + 41T^{2} \)
43 \( 1 + 0.476iT - 43T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 6.50iT - 59T^{2} \)
61 \( 1 - 8.25T + 61T^{2} \)
67 \( 1 - 3.29T + 67T^{2} \)
71 \( 1 + 4.29T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 6.07iT - 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + 11.6iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−8.604019692238117747335862235005, −8.314861573629112999050346791043, −7.11132455626617925771742377278, −5.87431890083204072343758350398, −5.44074155757670638812836850379, −4.74694010506052642082115972784, −4.07350435085095459617900229671, −3.35253657705025738665313317783, −2.10000692752627268519630864459, −0.856330373970348941542361002574, 0.908385312187481654795750914889, 1.88268413198603244958852108843, 2.50416615297604723842455644716, 3.70942198009965298804788571311, 4.53129576555385718527373999021, 5.59639827397702297291308335603, 6.59131928043845924838088015227, 6.81281982483750629209850831684, 7.68886378202280529947827298559, 7.910976355949878272258354336481

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