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  + 2.43·3-s + 1.30·5-s + 7-s + 2.92·9-s − 11-s − 13-s + 3.18·15-s + 5.30·17-s − 1.44·19-s + 2.43·21-s + 8.93·23-s − 3.28·25-s − 0.181·27-s − 5.58·29-s − 6.13·31-s − 2.43·33-s + 1.30·35-s + 8.97·37-s − 2.43·39-s − 3.64·41-s + 6.67·43-s + 3.83·45-s − 2.40·47-s + 49-s + 12.9·51-s + 4.61·53-s − 1.30·55-s + ⋯
L(s)  = 1  + 1.40·3-s + 0.585·5-s + 0.377·7-s + 0.975·9-s − 0.301·11-s − 0.277·13-s + 0.823·15-s + 1.28·17-s − 0.331·19-s + 0.531·21-s + 1.86·23-s − 0.657·25-s − 0.0349·27-s − 1.03·29-s − 1.10·31-s − 0.423·33-s + 0.221·35-s + 1.47·37-s − 0.389·39-s − 0.569·41-s + 1.01·43-s + 0.571·45-s − 0.351·47-s + 0.142·49-s + 1.80·51-s + 0.634·53-s − 0.176·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.349822959$
$L(\frac12)$  $\approx$  $4.349822959$
$L(\frac{3}{2})$   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 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 + 1.44T + 19T^{2} \)
23 \( 1 - 8.93T + 23T^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 - 8.97T + 37T^{2} \)
41 \( 1 + 3.64T + 41T^{2} \)
43 \( 1 - 6.67T + 43T^{2} \)
47 \( 1 + 2.40T + 47T^{2} \)
53 \( 1 - 4.61T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 0.118T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−7.73801466074942099998853064564, −7.49479718873950888148563518238, −6.60376652140844598259221341988, −5.51534272022517009038467995115, −5.21263282636098172801976288484, −4.00815060493492054520001230669, −3.46053202266129969278695975108, −2.55417936202686031582718895452, −2.05997796194135093907712092708, −0.997931210659447750062113828744, 0.997931210659447750062113828744, 2.05997796194135093907712092708, 2.55417936202686031582718895452, 3.46053202266129969278695975108, 4.00815060493492054520001230669, 5.21263282636098172801976288484, 5.51534272022517009038467995115, 6.60376652140844598259221341988, 7.49479718873950888148563518238, 7.73801466074942099998853064564

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