Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.29·3-s + 0.279·5-s − 7-s + 2.25·9-s + 11-s − 13-s + 0.639·15-s + 2.22·17-s − 8.55·19-s − 2.29·21-s − 3.73·23-s − 4.92·25-s − 1.70·27-s + 8.08·29-s − 8.68·31-s + 2.29·33-s − 0.279·35-s + 7.36·37-s − 2.29·39-s − 0.628·41-s − 2.32·43-s + 0.629·45-s − 3.73·47-s + 49-s + 5.09·51-s + 6.34·53-s + 0.279·55-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.124·5-s − 0.377·7-s + 0.751·9-s + 0.301·11-s − 0.277·13-s + 0.165·15-s + 0.539·17-s − 1.96·19-s − 0.500·21-s − 0.779·23-s − 0.984·25-s − 0.328·27-s + 1.50·29-s − 1.55·31-s + 0.399·33-s − 0.0471·35-s + 1.21·37-s − 0.367·39-s − 0.0980·41-s − 0.355·43-s + 0.0938·45-s − 0.545·47-s + 0.142·49-s + 0.713·51-s + 0.871·53-s + 0.0376·55-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  =  1
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 - 0.279T + 5T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 + 8.55T + 19T^{2} \)
23 \( 1 + 3.73T + 23T^{2} \)
29 \( 1 - 8.08T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 - 7.36T + 37T^{2} \)
41 \( 1 + 0.628T + 41T^{2} \)
43 \( 1 + 2.32T + 43T^{2} \)
47 \( 1 + 3.73T + 47T^{2} \)
53 \( 1 - 6.34T + 53T^{2} \)
59 \( 1 - 5.45T + 59T^{2} \)
61 \( 1 - 0.113T + 61T^{2} \)
67 \( 1 + 2.41T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 7.30T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 7.64T + 83T^{2} \)
89 \( 1 - 8.78T + 89T^{2} \)
97 \( 1 + 9.03T + 97T^{2} \)
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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

−7.73317002839494704614348776040, −6.85207732368259664086092088493, −6.20450054188187352962557253323, −5.48073324796106377090260132994, −4.25987908257716719872772649995, −3.95018861601973038390137557109, −2.98580043346744665196858015975, −2.34820392271169808717030094735, −1.59396526528548520263779594839, 0, 1.59396526528548520263779594839, 2.34820392271169808717030094735, 2.98580043346744665196858015975, 3.95018861601973038390137557109, 4.25987908257716719872772649995, 5.48073324796106377090260132994, 6.20450054188187352962557253323, 6.85207732368259664086092088493, 7.73317002839494704614348776040

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