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  + 1.46·3-s + 3.86·5-s − 7-s − 0.860·9-s − 11-s + 13-s + 5.64·15-s − 7.78·17-s − 4.18·19-s − 1.46·21-s − 9.11·23-s + 9.90·25-s − 5.64·27-s + 4.79·29-s − 8.32·31-s − 1.46·33-s − 3.86·35-s − 0.323·37-s + 1.46·39-s + 11.0·41-s − 2.58·43-s − 3.32·45-s + 8.36·47-s + 49-s − 11.3·51-s − 4.98·53-s − 3.86·55-s + ⋯
L(s)  = 1  + 0.844·3-s + 1.72·5-s − 0.377·7-s − 0.286·9-s − 0.301·11-s + 0.277·13-s + 1.45·15-s − 1.88·17-s − 0.959·19-s − 0.319·21-s − 1.90·23-s + 1.98·25-s − 1.08·27-s + 0.890·29-s − 1.49·31-s − 0.254·33-s − 0.652·35-s − 0.0531·37-s + 0.234·39-s + 1.72·41-s − 0.393·43-s − 0.495·45-s + 1.22·47-s + 0.142·49-s − 1.59·51-s − 0.685·53-s − 0.520·55-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 \( 1 - 1.46T + 3T^{2} \)
5 \( 1 - 3.86T + 5T^{2} \)
17 \( 1 + 7.78T + 17T^{2} \)
19 \( 1 + 4.18T + 19T^{2} \)
23 \( 1 + 9.11T + 23T^{2} \)
29 \( 1 - 4.79T + 29T^{2} \)
31 \( 1 + 8.32T + 31T^{2} \)
37 \( 1 + 0.323T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 - 8.36T + 47T^{2} \)
53 \( 1 + 4.98T + 53T^{2} \)
59 \( 1 + 4.97T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 - 1.18T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 1.44T + 73T^{2} \)
79 \( 1 + 2.33T + 79T^{2} \)
83 \( 1 + 6.98T + 83T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−7.51105983994662179256497527844, −6.64415420542117711965058257913, −6.02170469181912200130719556420, −5.71328393309283639644901701633, −4.57558867919856544924769673726, −3.88991272326361457793272365122, −2.74055194943842426213330460557, −2.29796990338717869773494490001, −1.71979115164282027571285931392, 0, 1.71979115164282027571285931392, 2.29796990338717869773494490001, 2.74055194943842426213330460557, 3.88991272326361457793272365122, 4.57558867919856544924769673726, 5.71328393309283639644901701633, 6.02170469181912200130719556420, 6.64415420542117711965058257913, 7.51105983994662179256497527844

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