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  + 1.69·3-s − 3.84·5-s − 7-s − 0.114·9-s − 11-s − 13-s − 6.52·15-s − 3.95·17-s − 0.644·19-s − 1.69·21-s + 9.08·23-s + 9.76·25-s − 5.29·27-s + 5.91·29-s − 1.51·31-s − 1.69·33-s + 3.84·35-s − 4.94·37-s − 1.69·39-s − 4.71·41-s + 6.12·43-s + 0.440·45-s − 11.9·47-s + 49-s − 6.71·51-s + 3.05·53-s + 3.84·55-s + ⋯
L(s)  = 1  + 0.980·3-s − 1.71·5-s − 0.377·7-s − 0.0382·9-s − 0.301·11-s − 0.277·13-s − 1.68·15-s − 0.959·17-s − 0.147·19-s − 0.370·21-s + 1.89·23-s + 1.95·25-s − 1.01·27-s + 1.09·29-s − 0.272·31-s − 0.295·33-s + 0.649·35-s − 0.812·37-s − 0.271·39-s − 0.736·41-s + 0.934·43-s + 0.0656·45-s − 1.74·47-s + 0.142·49-s − 0.940·51-s + 0.420·53-s + 0.518·55-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  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.275909735$
$L(\frac12)$  $\approx$  $1.275909735$
$L(\frac{3}{2})$   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 \( 1 - 1.69T + 3T^{2} \)
5 \( 1 + 3.84T + 5T^{2} \)
17 \( 1 + 3.95T + 17T^{2} \)
19 \( 1 + 0.644T + 19T^{2} \)
23 \( 1 - 9.08T + 23T^{2} \)
29 \( 1 - 5.91T + 29T^{2} \)
31 \( 1 + 1.51T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + 4.71T + 41T^{2} \)
43 \( 1 - 6.12T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 3.05T + 53T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + 7.58T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 8.90T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 8.07T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−8.293312842494329532765601271081, −7.984779041878348840307621001035, −7.01619440557207108707861523337, −6.67183492378270345586692085363, −5.18961603961769082074758352583, −4.55639932842581252373489954448, −3.58264365099289125703430149251, −3.16728857188006810621265246670, −2.27297012882301674633631324534, −0.58896781279062181126610340364, 0.58896781279062181126610340364, 2.27297012882301674633631324534, 3.16728857188006810621265246670, 3.58264365099289125703430149251, 4.55639932842581252373489954448, 5.18961603961769082074758352583, 6.67183492378270345586692085363, 7.01619440557207108707861523337, 7.984779041878348840307621001035, 8.293312842494329532765601271081

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