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  − 2.50·3-s − 3.40·5-s + 7-s + 3.27·9-s − 11-s − 13-s + 8.51·15-s + 4.74·17-s + 6.83·19-s − 2.50·21-s + 4.61·23-s + 6.56·25-s − 0.683·27-s + 5.06·29-s + 3.54·31-s + 2.50·33-s − 3.40·35-s − 6.49·37-s + 2.50·39-s − 1.94·41-s − 3.11·43-s − 11.1·45-s + 3.72·47-s + 49-s − 11.8·51-s + 2.20·53-s + 3.40·55-s + ⋯
L(s)  = 1  − 1.44·3-s − 1.52·5-s + 0.377·7-s + 1.09·9-s − 0.301·11-s − 0.277·13-s + 2.19·15-s + 1.15·17-s + 1.56·19-s − 0.546·21-s + 0.961·23-s + 1.31·25-s − 0.131·27-s + 0.940·29-s + 0.637·31-s + 0.435·33-s − 0.574·35-s − 1.06·37-s + 0.401·39-s − 0.303·41-s − 0.475·43-s − 1.65·45-s + 0.543·47-s + 0.142·49-s − 1.66·51-s + 0.302·53-s + 0.458·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8290095593$
$L(\frac12)$  $\approx$  $0.8290095593$
$L(\frac{3}{2})$   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 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
17 \( 1 - 4.74T + 17T^{2} \)
19 \( 1 - 6.83T + 19T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 - 5.06T + 29T^{2} \)
31 \( 1 - 3.54T + 31T^{2} \)
37 \( 1 + 6.49T + 37T^{2} \)
41 \( 1 + 1.94T + 41T^{2} \)
43 \( 1 + 3.11T + 43T^{2} \)
47 \( 1 - 3.72T + 47T^{2} \)
53 \( 1 - 2.20T + 53T^{2} \)
59 \( 1 + 7.10T + 59T^{2} \)
61 \( 1 - 3.05T + 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 - 2.08T + 73T^{2} \)
79 \( 1 + 8.96T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 - 3.09T + 97T^{2} \)
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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

−7.58266739228911749326727683971, −7.28163613310868988473367674172, −6.48078978666574772933507918963, −5.58080939104711471869376750798, −5.00543168672806217565459923511, −4.59536106745626822406642506857, −3.54503679265529292431471970059, −2.94685296956336963033932472468, −1.25935154167695522133222100457, −0.56642332168074100064339288061, 0.56642332168074100064339288061, 1.25935154167695522133222100457, 2.94685296956336963033932472468, 3.54503679265529292431471970059, 4.59536106745626822406642506857, 5.00543168672806217565459923511, 5.58080939104711471869376750798, 6.48078978666574772933507918963, 7.28163613310868988473367674172, 7.58266739228911749326727683971

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