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  + 1.15·3-s − 3.19·5-s − 7-s − 1.67·9-s + 11-s − 13-s − 3.67·15-s − 5.49·17-s − 5.11·19-s − 1.15·21-s + 1.48·23-s + 5.19·25-s − 5.38·27-s − 2.51·29-s − 7.49·31-s + 1.15·33-s + 3.19·35-s + 4.43·37-s − 1.15·39-s + 0.657·41-s + 6.68·43-s + 5.34·45-s + 2.89·47-s + 49-s − 6.32·51-s − 3.25·53-s − 3.19·55-s + ⋯
L(s)  = 1  + 0.664·3-s − 1.42·5-s − 0.377·7-s − 0.557·9-s + 0.301·11-s − 0.277·13-s − 0.949·15-s − 1.33·17-s − 1.17·19-s − 0.251·21-s + 0.310·23-s + 1.03·25-s − 1.03·27-s − 0.466·29-s − 1.34·31-s + 0.200·33-s + 0.539·35-s + 0.729·37-s − 0.184·39-s + 0.102·41-s + 1.01·43-s + 0.796·45-s + 0.422·47-s + 0.142·49-s − 0.885·51-s − 0.447·53-s − 0.430·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7176612532$
$L(\frac12)$  $\approx$  $0.7176612532$
$L(\frac{3}{2})$   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 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 + 3.19T + 5T^{2} \)
17 \( 1 + 5.49T + 17T^{2} \)
19 \( 1 + 5.11T + 19T^{2} \)
23 \( 1 - 1.48T + 23T^{2} \)
29 \( 1 + 2.51T + 29T^{2} \)
31 \( 1 + 7.49T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 - 0.657T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 + 3.25T + 53T^{2} \)
59 \( 1 - 3.56T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 7.93T + 67T^{2} \)
71 \( 1 - 6.97T + 71T^{2} \)
73 \( 1 + 5.60T + 73T^{2} \)
79 \( 1 - 8.60T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 9.04T + 89T^{2} \)
97 \( 1 - 7.78T + 97T^{2} \)
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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

−7.77484957626831727397767527176, −7.37380489569205929080096173267, −6.55570637099665458424065975798, −5.87539377417642127391244226065, −4.77685258046455056922267983474, −4.10314145363929521144913855006, −3.60431259344981631976559639380, −2.75345059090556870366203915664, −1.99712889425619858512111131672, −0.37938414777856202988291825376, 0.37938414777856202988291825376, 1.99712889425619858512111131672, 2.75345059090556870366203915664, 3.60431259344981631976559639380, 4.10314145363929521144913855006, 4.77685258046455056922267983474, 5.87539377417642127391244226065, 6.55570637099665458424065975798, 7.37380489569205929080096173267, 7.77484957626831727397767527176

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