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  − 2.16·3-s − 1.26·5-s − 7-s + 1.69·9-s − 11-s − 13-s + 2.74·15-s − 4.02·17-s − 5.92·19-s + 2.16·21-s + 4.27·23-s − 3.39·25-s + 2.82·27-s − 9.21·29-s − 0.386·31-s + 2.16·33-s + 1.26·35-s + 11.2·37-s + 2.16·39-s − 7.02·41-s − 3.45·43-s − 2.14·45-s − 9.82·47-s + 49-s + 8.71·51-s + 1.42·53-s + 1.26·55-s + ⋯
L(s)  = 1  − 1.25·3-s − 0.566·5-s − 0.377·7-s + 0.564·9-s − 0.301·11-s − 0.277·13-s + 0.708·15-s − 0.975·17-s − 1.36·19-s + 0.472·21-s + 0.891·23-s − 0.679·25-s + 0.544·27-s − 1.71·29-s − 0.0694·31-s + 0.377·33-s + 0.214·35-s + 1.84·37-s + 0.346·39-s − 1.09·41-s − 0.527·43-s − 0.319·45-s − 1.43·47-s + 0.142·49-s + 1.22·51-s + 0.195·53-s + 0.170·55-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  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2725517581$
$L(\frac12)$  $\approx$  $0.2725517581$
$L(\frac{3}{2})$   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 \( 1 + 2.16T + 3T^{2} \)
5 \( 1 + 1.26T + 5T^{2} \)
17 \( 1 + 4.02T + 17T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 - 4.27T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 + 0.386T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 7.02T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 + 9.82T + 47T^{2} \)
53 \( 1 - 1.42T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 0.110T + 71T^{2} \)
73 \( 1 + 3.03T + 73T^{2} \)
79 \( 1 + 0.962T + 79T^{2} \)
83 \( 1 - 4.40T + 83T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 - 3.14T + 97T^{2} \)
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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

−8.405132057028996237555631548745, −7.60642253347152808611748089606, −6.79891070590165180902639474767, −6.26938262847428544510590444438, −5.52386591521608668401680845769, −4.70799953237814711755752895821, −4.09150859706316966661777131265, −3.00216264774720184175353530707, −1.85910559747611358139286243764, −0.30440238716788046757474420969, 0.30440238716788046757474420969, 1.85910559747611358139286243764, 3.00216264774720184175353530707, 4.09150859706316966661777131265, 4.70799953237814711755752895821, 5.52386591521608668401680845769, 6.26938262847428544510590444438, 6.79891070590165180902639474767, 7.60642253347152808611748089606, 8.405132057028996237555631548745

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