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  − 2.52·3-s + 3.43·5-s + 7-s + 3.37·9-s − 11-s − 13-s − 8.68·15-s − 6.56·17-s + 2.74·19-s − 2.52·21-s − 2.80·23-s + 6.82·25-s − 0.947·27-s + 0.240·29-s + 0.281·31-s + 2.52·33-s + 3.43·35-s + 5.95·37-s + 2.52·39-s + 11.4·41-s − 5.59·43-s + 11.6·45-s − 13.3·47-s + 49-s + 16.5·51-s + 3.03·53-s − 3.43·55-s + ⋯
L(s)  = 1  − 1.45·3-s + 1.53·5-s + 0.377·7-s + 1.12·9-s − 0.301·11-s − 0.277·13-s − 2.24·15-s − 1.59·17-s + 0.629·19-s − 0.550·21-s − 0.584·23-s + 1.36·25-s − 0.182·27-s + 0.0446·29-s + 0.0505·31-s + 0.439·33-s + 0.581·35-s + 0.978·37-s + 0.404·39-s + 1.79·41-s − 0.853·43-s + 1.73·45-s − 1.94·47-s + 0.142·49-s + 2.32·51-s + 0.416·53-s − 0.463·55-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 \( 1 + 2.52T + 3T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
17 \( 1 + 6.56T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + 2.80T + 23T^{2} \)
29 \( 1 - 0.240T + 29T^{2} \)
31 \( 1 - 0.281T + 31T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 5.59T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 - 3.03T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 0.183T + 61T^{2} \)
67 \( 1 - 0.658T + 67T^{2} \)
71 \( 1 + 5.80T + 71T^{2} \)
73 \( 1 + 8.78T + 73T^{2} \)
79 \( 1 + 0.408T + 79T^{2} \)
83 \( 1 + 5.01T + 83T^{2} \)
89 \( 1 - 3.72T + 89T^{2} \)
97 \( 1 - 1.91T + 97T^{2} \)
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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

−7.25345987483765150148924193563, −6.46155555718940230719085830386, −6.10978380852407147084458744819, −5.49612158987922943698795414067, −4.84634392298320335137198481591, −4.34445902934550497958269136999, −2.86319275703723374856269747016, −2.05663774109585322202331003405, −1.24143660941215913007327243104, 0, 1.24143660941215913007327243104, 2.05663774109585322202331003405, 2.86319275703723374856269747016, 4.34445902934550497958269136999, 4.84634392298320335137198481591, 5.49612158987922943698795414067, 6.10978380852407147084458744819, 6.46155555718940230719085830386, 7.25345987483765150148924193563

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