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  + 0.293·3-s − 3.94·5-s + 7-s − 2.91·9-s − 11-s − 13-s − 1.15·15-s + 0.0585·17-s − 4.49·19-s + 0.293·21-s − 2.80·23-s + 10.5·25-s − 1.73·27-s − 7.01·29-s − 3.34·31-s − 0.293·33-s − 3.94·35-s − 9.50·37-s − 0.293·39-s − 9.75·41-s − 12.4·43-s + 11.4·45-s + 12.3·47-s + 49-s + 0.0171·51-s + 4.02·53-s + 3.94·55-s + ⋯
L(s)  = 1  + 0.169·3-s − 1.76·5-s + 0.377·7-s − 0.971·9-s − 0.301·11-s − 0.277·13-s − 0.298·15-s + 0.0142·17-s − 1.03·19-s + 0.0639·21-s − 0.584·23-s + 2.10·25-s − 0.333·27-s − 1.30·29-s − 0.600·31-s − 0.0510·33-s − 0.666·35-s − 1.56·37-s − 0.0469·39-s − 1.52·41-s − 1.89·43-s + 1.71·45-s + 1.79·47-s + 0.142·49-s + 0.00240·51-s + 0.553·53-s + 0.531·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3633837468$
$L(\frac12)$  $\approx$  $0.3633837468$
$L(\frac{3}{2})$   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 \( 1 - 0.293T + 3T^{2} \)
5 \( 1 + 3.94T + 5T^{2} \)
17 \( 1 - 0.0585T + 17T^{2} \)
19 \( 1 + 4.49T + 19T^{2} \)
23 \( 1 + 2.80T + 23T^{2} \)
29 \( 1 + 7.01T + 29T^{2} \)
31 \( 1 + 3.34T + 31T^{2} \)
37 \( 1 + 9.50T + 37T^{2} \)
41 \( 1 + 9.75T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 4.02T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 5.13T + 61T^{2} \)
67 \( 1 - 9.12T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 8.10T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 4.13T + 97T^{2} \)
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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

−8.012061063635619104724989594893, −7.15112488617780028165433769156, −6.76982597126631085845456054893, −5.48553747522486298664799583220, −5.13056323087275179995900165484, −3.93721307920865937044992452339, −3.79011844500480376859152741415, −2.78047353624407192800325537743, −1.85523199076702321991460043564, −0.28007954631653532598378553397, 0.28007954631653532598378553397, 1.85523199076702321991460043564, 2.78047353624407192800325537743, 3.79011844500480376859152741415, 3.93721307920865937044992452339, 5.13056323087275179995900165484, 5.48553747522486298664799583220, 6.76982597126631085845456054893, 7.15112488617780028165433769156, 8.012061063635619104724989594893

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