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  + 0.676·3-s + 3.59·5-s + 7-s − 2.54·9-s − 11-s − 13-s + 2.43·15-s − 5.62·17-s − 5.78·19-s + 0.676·21-s + 8.70·23-s + 7.91·25-s − 3.75·27-s + 9.58·29-s + 3.11·31-s − 0.676·33-s + 3.59·35-s − 5.39·37-s − 0.676·39-s + 8.94·41-s − 4.02·43-s − 9.13·45-s + 1.80·47-s + 49-s − 3.80·51-s − 4.67·53-s − 3.59·55-s + ⋯
L(s)  = 1  + 0.390·3-s + 1.60·5-s + 0.377·7-s − 0.847·9-s − 0.301·11-s − 0.277·13-s + 0.627·15-s − 1.36·17-s − 1.32·19-s + 0.147·21-s + 1.81·23-s + 1.58·25-s − 0.721·27-s + 1.78·29-s + 0.559·31-s − 0.117·33-s + 0.607·35-s − 0.887·37-s − 0.108·39-s + 1.39·41-s − 0.613·43-s − 1.36·45-s + 0.263·47-s + 0.142·49-s − 0.532·51-s − 0.641·53-s − 0.484·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.054166893$
$L(\frac12)$  $\approx$  $3.054166893$
$L(\frac{3}{2})$   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 \( 1 - 0.676T + 3T^{2} \)
5 \( 1 - 3.59T + 5T^{2} \)
17 \( 1 + 5.62T + 17T^{2} \)
19 \( 1 + 5.78T + 19T^{2} \)
23 \( 1 - 8.70T + 23T^{2} \)
29 \( 1 - 9.58T + 29T^{2} \)
31 \( 1 - 3.11T + 31T^{2} \)
37 \( 1 + 5.39T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 4.02T + 43T^{2} \)
47 \( 1 - 1.80T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 9.66T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 4.18T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 0.112T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−8.091829027891878835588921392268, −6.80508532888065616482643566770, −6.58849559490355438711395657728, −5.75504902704903939318200039736, −5.02127556791168040894988646543, −4.54903752567611196440294925515, −3.25211220989511351791809790879, −2.32778293813320108339055355589, −2.21022460218798287254113095578, −0.833969023280542130836075593919, 0.833969023280542130836075593919, 2.21022460218798287254113095578, 2.32778293813320108339055355589, 3.25211220989511351791809790879, 4.54903752567611196440294925515, 5.02127556791168040894988646543, 5.75504902704903939318200039736, 6.58849559490355438711395657728, 6.80508532888065616482643566770, 8.091829027891878835588921392268

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