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  + 3.30·3-s − 1.14·5-s − 7-s + 7.95·9-s − 11-s − 13-s − 3.77·15-s + 6.91·17-s − 0.199·19-s − 3.30·21-s + 3.36·23-s − 3.69·25-s + 16.3·27-s + 2.35·29-s − 2.74·31-s − 3.30·33-s + 1.14·35-s + 5.34·37-s − 3.30·39-s + 8.92·41-s − 1.44·43-s − 9.07·45-s − 0.453·47-s + 49-s + 22.8·51-s − 4.92·53-s + 1.14·55-s + ⋯
L(s)  = 1  + 1.91·3-s − 0.510·5-s − 0.377·7-s + 2.65·9-s − 0.301·11-s − 0.277·13-s − 0.975·15-s + 1.67·17-s − 0.0456·19-s − 0.722·21-s + 0.700·23-s − 0.739·25-s + 3.15·27-s + 0.436·29-s − 0.492·31-s − 0.576·33-s + 0.192·35-s + 0.878·37-s − 0.529·39-s + 1.39·41-s − 0.220·43-s − 1.35·45-s − 0.0661·47-s + 0.142·49-s + 3.20·51-s − 0.675·53-s + 0.153·55-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  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.737658800$
$L(\frac12)$  $\approx$  $3.737658800$
$L(\frac{3}{2})$   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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 + 1.14T + 5T^{2} \)
17 \( 1 - 6.91T + 17T^{2} \)
19 \( 1 + 0.199T + 19T^{2} \)
23 \( 1 - 3.36T + 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 2.74T + 31T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 + 0.453T + 47T^{2} \)
53 \( 1 + 4.92T + 53T^{2} \)
59 \( 1 + 2.39T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 7.69T + 71T^{2} \)
73 \( 1 - 9.77T + 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 - 1.96T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−8.327411721143866225837635727088, −7.73128464549458590502823601982, −7.45495871361793018815426661101, −6.48171960989691645622460473360, −5.34125027894193229549236764925, −4.33849820526570018865070105929, −3.61424844345391080088174793718, −3.03099715057741918116386375026, −2.26179818917661492741910081567, −1.06693675795323359981674547799, 1.06693675795323359981674547799, 2.26179818917661492741910081567, 3.03099715057741918116386375026, 3.61424844345391080088174793718, 4.33849820526570018865070105929, 5.34125027894193229549236764925, 6.48171960989691645622460473360, 7.45495871361793018815426661101, 7.73128464549458590502823601982, 8.327411721143866225837635727088

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