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  − 0.672·3-s + 0.538·5-s − 7-s − 2.54·9-s − 11-s − 13-s − 0.362·15-s − 1.70·17-s + 4.21·19-s + 0.672·21-s + 3.56·23-s − 4.71·25-s + 3.73·27-s + 9.32·29-s − 5.69·31-s + 0.672·33-s − 0.538·35-s − 8.10·37-s + 0.672·39-s − 6.73·41-s − 12.5·43-s − 1.37·45-s + 12.2·47-s + 49-s + 1.14·51-s + 7.52·53-s − 0.538·55-s + ⋯
L(s)  = 1  − 0.388·3-s + 0.240·5-s − 0.377·7-s − 0.849·9-s − 0.301·11-s − 0.277·13-s − 0.0935·15-s − 0.413·17-s + 0.967·19-s + 0.146·21-s + 0.742·23-s − 0.942·25-s + 0.718·27-s + 1.73·29-s − 1.02·31-s + 0.117·33-s − 0.0909·35-s − 1.33·37-s + 0.107·39-s − 1.05·41-s − 1.91·43-s − 0.204·45-s + 1.78·47-s + 0.142·49-s + 0.160·51-s + 1.03·53-s − 0.0725·55-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  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.180279298$
$L(\frac12)$  $\approx$  $1.180279298$
$L(\frac{3}{2})$   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 \( 1 + 0.672T + 3T^{2} \)
5 \( 1 - 0.538T + 5T^{2} \)
17 \( 1 + 1.70T + 17T^{2} \)
19 \( 1 - 4.21T + 19T^{2} \)
23 \( 1 - 3.56T + 23T^{2} \)
29 \( 1 - 9.32T + 29T^{2} \)
31 \( 1 + 5.69T + 31T^{2} \)
37 \( 1 + 8.10T + 37T^{2} \)
41 \( 1 + 6.73T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 7.52T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 - 8.27T + 61T^{2} \)
67 \( 1 + 8.26T + 67T^{2} \)
71 \( 1 - 9.58T + 71T^{2} \)
73 \( 1 - 7.50T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 0.0335T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−8.638859650495393610814522556876, −7.65064499858110231203072513392, −6.88405321360572885503827183526, −6.26643563318848736281868040893, −5.30289650359580125376482577982, −5.05153818184887872593791225853, −3.71732545550503442645150301343, −2.97832321458213724215339911632, −2.03452892552480394305352755768, −0.61380071038382942349273292698, 0.61380071038382942349273292698, 2.03452892552480394305352755768, 2.97832321458213724215339911632, 3.71732545550503442645150301343, 5.05153818184887872593791225853, 5.30289650359580125376482577982, 6.26643563318848736281868040893, 6.88405321360572885503827183526, 7.65064499858110231203072513392, 8.638859650495393610814522556876

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