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  + 1.59·3-s + 4.37·5-s − 7-s − 0.457·9-s − 11-s − 13-s + 6.96·15-s + 5.00·17-s + 5.82·19-s − 1.59·21-s + 4.91·23-s + 14.0·25-s − 5.51·27-s − 0.0873·29-s − 10.3·31-s − 1.59·33-s − 4.37·35-s − 4.36·37-s − 1.59·39-s + 3.46·41-s + 10.3·43-s − 2.00·45-s + 5.16·47-s + 49-s + 7.98·51-s + 9.50·53-s − 4.37·55-s + ⋯
L(s)  = 1  + 0.920·3-s + 1.95·5-s − 0.377·7-s − 0.152·9-s − 0.301·11-s − 0.277·13-s + 1.79·15-s + 1.21·17-s + 1.33·19-s − 0.347·21-s + 1.02·23-s + 2.81·25-s − 1.06·27-s − 0.0162·29-s − 1.85·31-s − 0.277·33-s − 0.738·35-s − 0.717·37-s − 0.255·39-s + 0.541·41-s + 1.57·43-s − 0.298·45-s + 0.752·47-s + 0.142·49-s + 1.11·51-s + 1.30·53-s − 0.589·55-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  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.842527716$
$L(\frac12)$  $\approx$  $3.842527716$
$L(\frac{3}{2})$   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 \( 1 - 1.59T + 3T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 + 0.0873T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 5.16T + 47T^{2} \)
53 \( 1 - 9.50T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 7.50T + 61T^{2} \)
67 \( 1 + 5.26T + 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 2.63T + 73T^{2} \)
79 \( 1 + 4.66T + 79T^{2} \)
83 \( 1 - 18.1T + 83T^{2} \)
89 \( 1 - 0.350T + 89T^{2} \)
97 \( 1 - 6.42T + 97T^{2} \)
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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

−8.800022357605693982244650650559, −7.52575625514002516020936511399, −7.24177588264114640313693211536, −5.88513613655159210130873099089, −5.71691899102391110861689936813, −4.90723289438651679245043983267, −3.42495265947108610722040986340, −2.89844054332735198535183219260, −2.13055483765513823235673811653, −1.15900776653224562826512213788, 1.15900776653224562826512213788, 2.13055483765513823235673811653, 2.89844054332735198535183219260, 3.42495265947108610722040986340, 4.90723289438651679245043983267, 5.71691899102391110861689936813, 5.88513613655159210130873099089, 7.24177588264114640313693211536, 7.52575625514002516020936511399, 8.800022357605693982244650650559

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