Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0962·3-s − 1.48·5-s − 7-s − 2.99·9-s − 11-s − 13-s − 0.143·15-s − 7.53·17-s + 7.59·19-s − 0.0962·21-s − 3.71·23-s − 2.78·25-s − 0.576·27-s − 9.29·29-s + 6.27·31-s − 0.0962·33-s + 1.48·35-s + 0.976·37-s − 0.0962·39-s + 4.24·41-s + 11.8·43-s + 4.44·45-s + 12.9·47-s + 49-s − 0.725·51-s − 4.71·53-s + 1.48·55-s + ⋯
L(s)  = 1  + 0.0555·3-s − 0.664·5-s − 0.377·7-s − 0.996·9-s − 0.301·11-s − 0.277·13-s − 0.0369·15-s − 1.82·17-s + 1.74·19-s − 0.0210·21-s − 0.775·23-s − 0.557·25-s − 0.110·27-s − 1.72·29-s + 1.12·31-s − 0.0167·33-s + 0.251·35-s + 0.160·37-s − 0.0154·39-s + 0.662·41-s + 1.81·43-s + 0.662·45-s + 1.88·47-s + 0.142·49-s − 0.101·51-s − 0.648·53-s + 0.200·55-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

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8803946670$
$L(\frac12)$  $\approx$  $0.8803946670$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 - 0.0962T + 3T^{2} \)
5 \( 1 + 1.48T + 5T^{2} \)
17 \( 1 + 7.53T + 17T^{2} \)
19 \( 1 - 7.59T + 19T^{2} \)
23 \( 1 + 3.71T + 23T^{2} \)
29 \( 1 + 9.29T + 29T^{2} \)
31 \( 1 - 6.27T + 31T^{2} \)
37 \( 1 - 0.976T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 + 5.01T + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 - 5.47T + 67T^{2} \)
71 \( 1 + 9.99T + 71T^{2} \)
73 \( 1 - 8.02T + 73T^{2} \)
79 \( 1 - 2.16T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 9.20T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.405262058766613277080694153114, −7.64228244583136364433109462957, −7.21568689721905653050509523476, −6.07571809416931066782626888975, −5.63904454724782136296871449610, −4.56136052226701392980220233384, −3.86023988242690875911140821139, −2.92873904115897717827762287804, −2.18343151295769478353011713357, −0.50556033794814371888545962170, 0.50556033794814371888545962170, 2.18343151295769478353011713357, 2.92873904115897717827762287804, 3.86023988242690875911140821139, 4.56136052226701392980220233384, 5.63904454724782136296871449610, 6.07571809416931066782626888975, 7.21568689721905653050509523476, 7.64228244583136364433109462957, 8.405262058766613277080694153114

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