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  − 1.80·3-s − 2.97·5-s − 7-s + 0.244·9-s − 11-s − 13-s + 5.36·15-s − 3.36·17-s − 0.650·19-s + 1.80·21-s − 5.75·23-s + 3.87·25-s + 4.96·27-s + 3.84·29-s + 4.81·31-s + 1.80·33-s + 2.97·35-s + 5.38·37-s + 1.80·39-s + 0.331·41-s − 4.36·43-s − 0.729·45-s + 10.3·47-s + 49-s + 6.05·51-s + 2.85·53-s + 2.97·55-s + ⋯
L(s)  = 1  − 1.04·3-s − 1.33·5-s − 0.377·7-s + 0.0816·9-s − 0.301·11-s − 0.277·13-s + 1.38·15-s − 0.815·17-s − 0.149·19-s + 0.393·21-s − 1.19·23-s + 0.774·25-s + 0.955·27-s + 0.713·29-s + 0.864·31-s + 0.313·33-s + 0.503·35-s + 0.885·37-s + 0.288·39-s + 0.0517·41-s − 0.664·43-s − 0.108·45-s + 1.50·47-s + 0.142·49-s + 0.848·51-s + 0.391·53-s + 0.401·55-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 \( 1 + 1.80T + 3T^{2} \)
5 \( 1 + 2.97T + 5T^{2} \)
17 \( 1 + 3.36T + 17T^{2} \)
19 \( 1 + 0.650T + 19T^{2} \)
23 \( 1 + 5.75T + 23T^{2} \)
29 \( 1 - 3.84T + 29T^{2} \)
31 \( 1 - 4.81T + 31T^{2} \)
37 \( 1 - 5.38T + 37T^{2} \)
41 \( 1 - 0.331T + 41T^{2} \)
43 \( 1 + 4.36T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 2.85T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 8.41T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 + 7.46T + 97T^{2} \)
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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

−7.40363572700458317785465970129, −6.76592482521518772662691186315, −6.11763614195934124989005822280, −5.46208979606369993923678366565, −4.49177894661903013495552625942, −4.20523273844317512181657047303, −3.16255517958195757800540197823, −2.32497437985182095910618949073, −0.77607278394989033482329017714, 0, 0.77607278394989033482329017714, 2.32497437985182095910618949073, 3.16255517958195757800540197823, 4.20523273844317512181657047303, 4.49177894661903013495552625942, 5.46208979606369993923678366565, 6.11763614195934124989005822280, 6.76592482521518772662691186315, 7.40363572700458317785465970129

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