Properties

Label 2-4004-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95·3-s − 0.209·5-s − 7-s + 0.827·9-s + 11-s − 13-s + 0.408·15-s + 2.88·17-s − 1.89·19-s + 1.95·21-s − 1.33·23-s − 4.95·25-s + 4.25·27-s + 5.48·29-s + 4.65·31-s − 1.95·33-s + 0.209·35-s + 1.56·37-s + 1.95·39-s − 7.25·41-s + 4.32·43-s − 0.172·45-s + 3.41·47-s + 49-s − 5.64·51-s − 10.2·53-s − 0.209·55-s + ⋯
L(s)  = 1  − 1.12·3-s − 0.0934·5-s − 0.377·7-s + 0.275·9-s + 0.301·11-s − 0.277·13-s + 0.105·15-s + 0.699·17-s − 0.435·19-s + 0.426·21-s − 0.279·23-s − 0.991·25-s + 0.818·27-s + 1.01·29-s + 0.835·31-s − 0.340·33-s + 0.0353·35-s + 0.258·37-s + 0.313·39-s − 1.13·41-s + 0.659·43-s − 0.0257·45-s + 0.497·47-s + 0.142·49-s − 0.790·51-s − 1.40·53-s − 0.0281·55-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: \(1\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.95T + 3T^{2} \)
5 \( 1 + 0.209T + 5T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 + 1.89T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 - 5.48T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 - 1.56T + 37T^{2} \)
41 \( 1 + 7.25T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 + 0.657T + 61T^{2} \)
67 \( 1 - 2.30T + 67T^{2} \)
71 \( 1 + 8.47T + 71T^{2} \)
73 \( 1 + 4.90T + 73T^{2} \)
79 \( 1 - 7.52T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−8.055966220084498617764817631956, −7.21519324940920901095019987591, −6.37920073912689053281199461394, −6.00263703759491526335786276763, −5.14503444369706779597684160915, −4.43525450851473754373342687096, −3.49227547180444923993031098197, −2.47490101735427255017314982929, −1.13940660503600657452161457975, 0, 1.13940660503600657452161457975, 2.47490101735427255017314982929, 3.49227547180444923993031098197, 4.43525450851473754373342687096, 5.14503444369706779597684160915, 6.00263703759491526335786276763, 6.37920073912689053281199461394, 7.21519324940920901095019987591, 8.055966220084498617764817631956

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