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  + 2.29·3-s + 3.91·5-s − 7-s + 2.25·9-s + 11-s − 13-s + 8.97·15-s + 5.29·17-s + 2.52·19-s − 2.29·21-s − 4.13·23-s + 10.3·25-s − 1.70·27-s + 6.65·29-s − 3.98·31-s + 2.29·33-s − 3.91·35-s + 2.57·37-s − 2.29·39-s + 4.72·41-s − 7.81·43-s + 8.82·45-s + 0.823·47-s + 49-s + 12.1·51-s + 0.474·53-s + 3.91·55-s + ⋯
L(s)  = 1  + 1.32·3-s + 1.75·5-s − 0.377·7-s + 0.751·9-s + 0.301·11-s − 0.277·13-s + 2.31·15-s + 1.28·17-s + 0.579·19-s − 0.500·21-s − 0.861·23-s + 2.06·25-s − 0.328·27-s + 1.23·29-s − 0.715·31-s + 0.399·33-s − 0.661·35-s + 0.423·37-s − 0.367·39-s + 0.737·41-s − 1.19·43-s + 1.31·45-s + 0.120·47-s + 0.142·49-s + 1.69·51-s + 0.0651·53-s + 0.528·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.067888719$
$L(\frac12)$  $\approx$  $5.067888719$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 - 3.91T + 5T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 2.52T + 19T^{2} \)
23 \( 1 + 4.13T + 23T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + 3.98T + 31T^{2} \)
37 \( 1 - 2.57T + 37T^{2} \)
41 \( 1 - 4.72T + 41T^{2} \)
43 \( 1 + 7.81T + 43T^{2} \)
47 \( 1 - 0.823T + 47T^{2} \)
53 \( 1 - 0.474T + 53T^{2} \)
59 \( 1 - 9.39T + 59T^{2} \)
61 \( 1 - 6.43T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 1.43T + 97T^{2} \)
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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

−8.020249816265219784523060571899, −7.14066421039395044284246894453, −6.48838119178765581944527705013, −5.69670594487850162837065138879, −5.24926478092810691575214138562, −4.09318965785764381139155149101, −3.24845646739175221189469683537, −2.65188517407452467510325753386, −1.96697049031867533096641116206, −1.12030375127877512008095934409, 1.12030375127877512008095934409, 1.96697049031867533096641116206, 2.65188517407452467510325753386, 3.24845646739175221189469683537, 4.09318965785764381139155149101, 5.24926478092810691575214138562, 5.69670594487850162837065138879, 6.48838119178765581944527705013, 7.14066421039395044284246894453, 8.020249816265219784523060571899

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