Properties

Degree 2
Conductor $ 2^{3} \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  − 3.07·3-s − 3.53·5-s − 7-s + 6.42·9-s + 11-s − 13-s + 10.8·15-s + 1.83·17-s − 0.318·19-s + 3.07·21-s + 1.10·23-s + 7.53·25-s − 10.5·27-s − 1.36·29-s − 1.34·31-s − 3.07·33-s + 3.53·35-s + 10.6·37-s + 3.07·39-s + 0.852·41-s − 2.13·43-s − 22.7·45-s + 10.5·47-s + 49-s − 5.63·51-s − 1.45·53-s − 3.53·55-s + ⋯
L(s)  = 1  − 1.77·3-s − 1.58·5-s − 0.377·7-s + 2.14·9-s + 0.301·11-s − 0.277·13-s + 2.80·15-s + 0.445·17-s − 0.0729·19-s + 0.670·21-s + 0.230·23-s + 1.50·25-s − 2.02·27-s − 0.252·29-s − 0.241·31-s − 0.534·33-s + 0.598·35-s + 1.75·37-s + 0.491·39-s + 0.133·41-s − 0.326·43-s − 3.39·45-s + 1.53·47-s + 0.142·49-s − 0.789·51-s − 0.199·53-s − 0.477·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4336640000$
$L(\frac12)$  $\approx$  $0.4336640000$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + 3.07T + 3T^{2} \)
5 \( 1 + 3.53T + 5T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 + 0.318T + 19T^{2} \)
23 \( 1 - 1.10T + 23T^{2} \)
29 \( 1 + 1.36T + 29T^{2} \)
31 \( 1 + 1.34T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 0.852T + 41T^{2} \)
43 \( 1 + 2.13T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 1.45T + 53T^{2} \)
59 \( 1 + 5.50T + 59T^{2} \)
61 \( 1 + 4.79T + 61T^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 - 0.673T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + 6.33T + 79T^{2} \)
83 \( 1 + 4.51T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 3.08T + 97T^{2} \)
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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

−7.52938690042910109002251476224, −7.17617144046848386464100849437, −6.40732025799591846260330755293, −5.79184368491118776972458331621, −5.03636667417531940615898440087, −4.29575759022963626706274828599, −3.88494329760799009299835010364, −2.82522564737491086171870782116, −1.26138472878929005896051748070, −0.40772237337830975635546737568, 0.40772237337830975635546737568, 1.26138472878929005896051748070, 2.82522564737491086171870782116, 3.88494329760799009299835010364, 4.29575759022963626706274828599, 5.03636667417531940615898440087, 5.79184368491118776972458331621, 6.40732025799591846260330755293, 7.17617144046848386464100849437, 7.52938690042910109002251476224

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