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  + 0.674·3-s − 4.42·5-s − 7-s − 2.54·9-s + 11-s − 13-s − 2.98·15-s + 4.81·17-s − 5.29·19-s − 0.674·21-s + 1.44·23-s + 14.5·25-s − 3.74·27-s + 7.93·29-s + 8.10·31-s + 0.674·33-s + 4.42·35-s − 6.99·37-s − 0.674·39-s + 7.05·41-s − 7.92·43-s + 11.2·45-s + 1.44·47-s + 49-s + 3.25·51-s − 8.55·53-s − 4.42·55-s + ⋯
L(s)  = 1  + 0.389·3-s − 1.97·5-s − 0.377·7-s − 0.848·9-s + 0.301·11-s − 0.277·13-s − 0.770·15-s + 1.16·17-s − 1.21·19-s − 0.147·21-s + 0.302·23-s + 2.91·25-s − 0.720·27-s + 1.47·29-s + 1.45·31-s + 0.117·33-s + 0.747·35-s − 1.15·37-s − 0.108·39-s + 1.10·41-s − 1.20·43-s + 1.67·45-s + 0.211·47-s + 0.142·49-s + 0.455·51-s − 1.17·53-s − 0.596·55-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 \( 1 - 0.674T + 3T^{2} \)
5 \( 1 + 4.42T + 5T^{2} \)
17 \( 1 - 4.81T + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 - 1.44T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 - 8.10T + 31T^{2} \)
37 \( 1 + 6.99T + 37T^{2} \)
41 \( 1 - 7.05T + 41T^{2} \)
43 \( 1 + 7.92T + 43T^{2} \)
47 \( 1 - 1.44T + 47T^{2} \)
53 \( 1 + 8.55T + 53T^{2} \)
59 \( 1 + 0.0353T + 59T^{2} \)
61 \( 1 - 9.90T + 61T^{2} \)
67 \( 1 + 1.89T + 67T^{2} \)
71 \( 1 - 9.06T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 9.53T + 83T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−7.71477936474994147544028262148, −6.81457097074801395362958321016, −6.36877331505546045059457065131, −5.20664283068982938670405706711, −4.53103258612177207826459550413, −3.78054676226467346678582352814, −3.20039773924253852443454490890, −2.57886791464118657542958795947, −0.982913487393907003824736734847, 0, 0.982913487393907003824736734847, 2.57886791464118657542958795947, 3.20039773924253852443454490890, 3.78054676226467346678582352814, 4.53103258612177207826459550413, 5.20664283068982938670405706711, 6.36877331505546045059457065131, 6.81457097074801395362958321016, 7.71477936474994147544028262148

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