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  − 0.333·3-s + 1.97·5-s − 7-s − 2.88·9-s + 11-s + 13-s − 0.658·15-s + 3.01·17-s + 7.23·19-s + 0.333·21-s − 5.73·23-s − 1.10·25-s + 1.96·27-s − 5.03·29-s − 7.65·31-s − 0.333·33-s − 1.97·35-s + 2.20·37-s − 0.333·39-s − 7.09·41-s + 8.24·43-s − 5.69·45-s − 6.28·47-s + 49-s − 1.00·51-s − 13.7·53-s + 1.97·55-s + ⋯
L(s)  = 1  − 0.192·3-s + 0.882·5-s − 0.377·7-s − 0.962·9-s + 0.301·11-s + 0.277·13-s − 0.170·15-s + 0.730·17-s + 1.65·19-s + 0.0728·21-s − 1.19·23-s − 0.221·25-s + 0.378·27-s − 0.935·29-s − 1.37·31-s − 0.0580·33-s − 0.333·35-s + 0.362·37-s − 0.0534·39-s − 1.10·41-s + 1.25·43-s − 0.849·45-s − 0.916·47-s + 0.142·49-s − 0.140·51-s − 1.88·53-s + 0.266·55-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 \( 1 + 0.333T + 3T^{2} \)
5 \( 1 - 1.97T + 5T^{2} \)
17 \( 1 - 3.01T + 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 + 7.65T + 31T^{2} \)
37 \( 1 - 2.20T + 37T^{2} \)
41 \( 1 + 7.09T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 6.71T + 61T^{2} \)
67 \( 1 + 0.389T + 67T^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 9.98T + 79T^{2} \)
83 \( 1 - 4.15T + 83T^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−7.64885723464392844036993747380, −6.59884963054282391168498732104, −5.95946174718665467272575052751, −5.59887204939066807616764049316, −4.90613763584161065337716079202, −3.63716113349265151342764970650, −3.23470863993461916772298950456, −2.16063502193691594456283304811, −1.33958762339205020865452462965, 0, 1.33958762339205020865452462965, 2.16063502193691594456283304811, 3.23470863993461916772298950456, 3.63716113349265151342764970650, 4.90613763584161065337716079202, 5.59887204939066807616764049316, 5.95946174718665467272575052751, 6.59884963054282391168498732104, 7.64885723464392844036993747380

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