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  − 2.44·3-s + 0.430·5-s − 7-s + 2.98·9-s − 11-s − 13-s − 1.05·15-s − 2.46·17-s + 2.25·19-s + 2.44·21-s + 0.471·23-s − 4.81·25-s + 0.0427·27-s + 3.50·29-s + 2.03·31-s + 2.44·33-s − 0.430·35-s − 3.29·37-s + 2.44·39-s + 4.77·41-s + 6.41·43-s + 1.28·45-s + 4.46·47-s + 49-s + 6.01·51-s − 14.0·53-s − 0.430·55-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.192·5-s − 0.377·7-s + 0.994·9-s − 0.301·11-s − 0.277·13-s − 0.271·15-s − 0.596·17-s + 0.517·19-s + 0.533·21-s + 0.0984·23-s − 0.962·25-s + 0.00823·27-s + 0.651·29-s + 0.365·31-s + 0.425·33-s − 0.0727·35-s − 0.542·37-s + 0.391·39-s + 0.746·41-s + 0.978·43-s + 0.191·45-s + 0.650·47-s + 0.142·49-s + 0.842·51-s − 1.93·53-s − 0.0580·55-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 \( 1 + 2.44T + 3T^{2} \)
5 \( 1 - 0.430T + 5T^{2} \)
17 \( 1 + 2.46T + 17T^{2} \)
19 \( 1 - 2.25T + 19T^{2} \)
23 \( 1 - 0.471T + 23T^{2} \)
29 \( 1 - 3.50T + 29T^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 + 3.29T + 37T^{2} \)
41 \( 1 - 4.77T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 - 4.46T + 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 - 2.10T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 8.00T + 73T^{2} \)
79 \( 1 - 7.77T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 - 9.75T + 97T^{2} \)
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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

−7.40524711527355295844410611026, −6.48631819792573697205118500419, −6.20634339615402105354963862745, −5.41732316780378720461528748854, −4.85935802948858839776328463156, −4.12889577171491187115594656444, −3.09036117504363757515978479090, −2.15449568593450195918357390586, −0.972757017434179077151144494539, 0, 0.972757017434179077151144494539, 2.15449568593450195918357390586, 3.09036117504363757515978479090, 4.12889577171491187115594656444, 4.85935802948858839776328463156, 5.41732316780378720461528748854, 6.20634339615402105354963862745, 6.48631819792573697205118500419, 7.40524711527355295844410611026

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