Properties

Label 2-4004-13.12-c1-0-44
Degree $2$
Conductor $4004$
Sign $-0.814 + 0.580i$
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  − 1.49·3-s − 1.37i·5-s + i·7-s − 0.776·9-s + i·11-s + (−2.93 + 2.09i)13-s + 2.05i·15-s + 0.801·17-s − 0.0452i·19-s − 1.49i·21-s + 6.18·23-s + 3.09·25-s + 5.63·27-s − 7.84·29-s + 2.31i·31-s + ⋯
L(s)  = 1  − 0.860·3-s − 0.616i·5-s + 0.377i·7-s − 0.258·9-s + 0.301i·11-s + (−0.814 + 0.580i)13-s + 0.530i·15-s + 0.194·17-s − 0.0103i·19-s − 0.325i·21-s + 1.29·23-s + 0.619·25-s + 1.08·27-s − 1.45·29-s + 0.415i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ -0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2947841554\)
\(L(\frac12)\) \(\approx\) \(0.2947841554\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
11 \( 1 - iT \)
13 \( 1 + (2.93 - 2.09i)T \)
good3 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 + 1.37iT - 5T^{2} \)
17 \( 1 - 0.801T + 17T^{2} \)
19 \( 1 + 0.0452iT - 19T^{2} \)
23 \( 1 - 6.18T + 23T^{2} \)
29 \( 1 + 7.84T + 29T^{2} \)
31 \( 1 - 2.31iT - 31T^{2} \)
37 \( 1 - 3.74iT - 37T^{2} \)
41 \( 1 + 2.15iT - 41T^{2} \)
43 \( 1 + 4.55T + 43T^{2} \)
47 \( 1 + 1.91iT - 47T^{2} \)
53 \( 1 - 1.32T + 53T^{2} \)
59 \( 1 + 8.56iT - 59T^{2} \)
61 \( 1 - 7.82T + 61T^{2} \)
67 \( 1 + 13.0iT - 67T^{2} \)
71 \( 1 - 1.36iT - 71T^{2} \)
73 \( 1 - 6.13iT - 73T^{2} \)
79 \( 1 + 8.88T + 79T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 + 9.55iT - 89T^{2} \)
97 \( 1 + 5.59iT - 97T^{2} \)
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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

−8.265786236476556399412199237414, −7.22878224894064328560715795574, −6.71147242286008199166989362965, −5.79480033754149987579268861269, −5.04316370619114537452642151466, −4.79085572233594116325232146516, −3.53263267605875061782440769837, −2.50158385399594058713860829691, −1.38656598647225187028721270466, −0.11279152575154912445327550592, 1.03507530715521609773207958220, 2.54319799900918159860890906502, 3.22549365077890596176950980882, 4.26132650974639190590734615274, 5.29225017716888373490550931638, 5.60540497251952153382611506365, 6.63080676193911000923184493607, 7.12295483125242381866886386731, 7.85456287017832552784831790915, 8.749870684810494167296799866298

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