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  + 2.97·3-s − 3.07·5-s − 7-s + 5.83·9-s − 11-s − 13-s − 9.14·15-s + 2.23·17-s + 0.103·19-s − 2.97·21-s − 2.91·23-s + 4.47·25-s + 8.42·27-s + 5.61·29-s − 0.191·31-s − 2.97·33-s + 3.07·35-s − 6.04·37-s − 2.97·39-s − 10.4·41-s + 4.42·43-s − 17.9·45-s − 4.29·47-s + 49-s + 6.64·51-s − 0.962·53-s + 3.07·55-s + ⋯
L(s)  = 1  + 1.71·3-s − 1.37·5-s − 0.377·7-s + 1.94·9-s − 0.301·11-s − 0.277·13-s − 2.36·15-s + 0.542·17-s + 0.0237·19-s − 0.648·21-s − 0.608·23-s + 0.894·25-s + 1.62·27-s + 1.04·29-s − 0.0344·31-s − 0.517·33-s + 0.520·35-s − 0.994·37-s − 0.475·39-s − 1.63·41-s + 0.675·43-s − 2.67·45-s − 0.626·47-s + 0.142·49-s + 0.930·51-s − 0.132·53-s + 0.415·55-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 \( 1 - 2.97T + 3T^{2} \)
5 \( 1 + 3.07T + 5T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 - 0.103T + 19T^{2} \)
23 \( 1 + 2.91T + 23T^{2} \)
29 \( 1 - 5.61T + 29T^{2} \)
31 \( 1 + 0.191T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 4.42T + 43T^{2} \)
47 \( 1 + 4.29T + 47T^{2} \)
53 \( 1 + 0.962T + 53T^{2} \)
59 \( 1 - 7.41T + 59T^{2} \)
61 \( 1 + 6.36T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 0.646T + 71T^{2} \)
73 \( 1 - 2.81T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 - 1.79T + 83T^{2} \)
89 \( 1 - 2.92T + 89T^{2} \)
97 \( 1 + 1.39T + 97T^{2} \)
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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

−7.55345197403233134623999837107, −7.22022814204802927348432008479, −6.35972292367315124348624545741, −5.14818720410772832414701027003, −4.35293693765477546339774568668, −3.69296829550366233626710298658, −3.19418505371999259952969212958, −2.51517401603686196991246059394, −1.43764328632949055633781768952, 0, 1.43764328632949055633781768952, 2.51517401603686196991246059394, 3.19418505371999259952969212958, 3.69296829550366233626710298658, 4.35293693765477546339774568668, 5.14818720410772832414701027003, 6.35972292367315124348624545741, 7.22022814204802927348432008479, 7.55345197403233134623999837107

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