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  + 0.0822·3-s − 4.16·5-s − 7-s − 2.99·9-s − 11-s − 13-s − 0.342·15-s + 3.55·17-s + 5.09·19-s − 0.0822·21-s − 0.867·23-s + 12.3·25-s − 0.492·27-s − 8.23·29-s + 2.28·31-s − 0.0822·33-s + 4.16·35-s + 6.74·37-s − 0.0822·39-s + 2.15·41-s + 6.41·43-s + 12.4·45-s − 13.3·47-s + 49-s + 0.292·51-s − 2.36·53-s + 4.16·55-s + ⋯
L(s)  = 1  + 0.0474·3-s − 1.86·5-s − 0.377·7-s − 0.997·9-s − 0.301·11-s − 0.277·13-s − 0.0884·15-s + 0.863·17-s + 1.16·19-s − 0.0179·21-s − 0.180·23-s + 2.47·25-s − 0.0948·27-s − 1.52·29-s + 0.410·31-s − 0.0143·33-s + 0.704·35-s + 1.10·37-s − 0.0131·39-s + 0.336·41-s + 0.978·43-s + 1.85·45-s − 1.94·47-s + 0.142·49-s + 0.0409·51-s − 0.324·53-s + 0.561·55-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 \( 1 - 0.0822T + 3T^{2} \)
5 \( 1 + 4.16T + 5T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 - 5.09T + 19T^{2} \)
23 \( 1 + 0.867T + 23T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 - 6.74T + 37T^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 + 2.36T + 53T^{2} \)
59 \( 1 - 2.85T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 - 1.56T + 71T^{2} \)
73 \( 1 - 4.34T + 73T^{2} \)
79 \( 1 - 0.0822T + 79T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 - 5.43T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−7.72618694905280351932706839884, −7.02684571013490978346859200870, −6.07476586076651330305426925491, −5.33492430135165461647354519577, −4.63780527731195108694955509466, −3.65270290589470474259906183677, −3.33410815836188963593938507222, −2.50986716267567213031602799535, −0.914648945021387609450356470762, 0, 0.914648945021387609450356470762, 2.50986716267567213031602799535, 3.33410815836188963593938507222, 3.65270290589470474259906183677, 4.63780527731195108694955509466, 5.33492430135165461647354519577, 6.07476586076651330305426925491, 7.02684571013490978346859200870, 7.72618694905280351932706839884

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