Properties

Label 2-4004-77.76-c1-0-69
Degree $2$
Conductor $4004$
Sign $0.907 + 0.419i$
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  + 0.153i·3-s + 2.94i·5-s + (0.319 − 2.62i)7-s + 2.97·9-s + (1.01 − 3.15i)11-s − 13-s − 0.452·15-s + 5.02·17-s + 4.07·19-s + (0.403 + 0.0490i)21-s − 6.95·23-s − 3.69·25-s + 0.917i·27-s − 5.04i·29-s + 0.575i·31-s + ⋯
L(s)  = 1  + 0.0886i·3-s + 1.31i·5-s + (0.120 − 0.992i)7-s + 0.992·9-s + (0.306 − 0.951i)11-s − 0.277·13-s − 0.116·15-s + 1.21·17-s + 0.935·19-s + (0.0879 + 0.0107i)21-s − 1.45·23-s − 0.739·25-s + 0.176i·27-s − 0.936i·29-s + 0.103i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.907 + 0.419i$
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: \((2,\ 4004,\ (\ :1/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.149463997\)
\(L(\frac12)\) \(\approx\) \(2.149463997\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.319 + 2.62i)T \)
11 \( 1 + (-1.01 + 3.15i)T \)
13 \( 1 + T \)
good3 \( 1 - 0.153iT - 3T^{2} \)
5 \( 1 - 2.94iT - 5T^{2} \)
17 \( 1 - 5.02T + 17T^{2} \)
19 \( 1 - 4.07T + 19T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 + 5.04iT - 29T^{2} \)
31 \( 1 - 0.575iT - 31T^{2} \)
37 \( 1 - 3.52T + 37T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 + 5.12iT - 43T^{2} \)
47 \( 1 + 6.71iT - 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + 13.5iT - 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 0.423T + 73T^{2} \)
79 \( 1 + 8.71iT - 79T^{2} \)
83 \( 1 + 9.18T + 83T^{2} \)
89 \( 1 - 4.93iT - 89T^{2} \)
97 \( 1 + 0.793iT - 97T^{2} \)
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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

−8.015447387062611270511758444425, −7.67416905195310719879513605785, −6.98942929248234386767279966268, −6.31132135528321599797455054653, −5.56090234087103249353966401306, −4.41218375651125178934985371592, −3.65837979281497524316909655890, −3.14349090052851226288921060816, −1.89000515791550366639579376114, −0.70767018253645977588760301893, 1.15752025238813899975728594338, 1.76241880162342491857377847904, 2.95596621562629160644189897972, 4.19811769868418736090694043182, 4.69313185123313704826582507792, 5.48739129638872827715104202654, 6.09032847787431600777221790521, 7.26751403316094255161632737191, 7.76407508760190956775804372610, 8.484182877770742968304021824136

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