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  + 1.62·3-s + 1.90·5-s − 7-s − 0.354·9-s + 11-s − 13-s + 3.10·15-s − 3.20·17-s − 1.68·19-s − 1.62·21-s − 6.28·23-s − 1.35·25-s − 5.45·27-s − 10.4·29-s + 10.9·31-s + 1.62·33-s − 1.90·35-s + 7.50·37-s − 1.62·39-s + 4.62·41-s + 6.59·43-s − 0.676·45-s − 6.28·47-s + 49-s − 5.21·51-s − 0.328·53-s + 1.90·55-s + ⋯
L(s)  = 1  + 0.939·3-s + 0.853·5-s − 0.377·7-s − 0.118·9-s + 0.301·11-s − 0.277·13-s + 0.801·15-s − 0.778·17-s − 0.385·19-s − 0.354·21-s − 1.31·23-s − 0.271·25-s − 1.05·27-s − 1.93·29-s + 1.97·31-s + 0.283·33-s − 0.322·35-s + 1.23·37-s − 0.260·39-s + 0.722·41-s + 1.00·43-s − 0.100·45-s − 0.917·47-s + 0.142·49-s − 0.730·51-s − 0.0451·53-s + 0.257·55-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 \( 1 - 1.62T + 3T^{2} \)
5 \( 1 - 1.90T + 5T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 + 1.68T + 19T^{2} \)
23 \( 1 + 6.28T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 - 7.50T + 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 - 6.59T + 43T^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 + 0.328T + 53T^{2} \)
59 \( 1 + 0.221T + 59T^{2} \)
61 \( 1 + 2.97T + 61T^{2} \)
67 \( 1 - 5.06T + 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 8.26T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−7.66889360904242090685001518418, −6.76439742521779500910512964545, −5.99483726740740531161503807902, −5.69680529799440054433658676649, −4.40511957590511309406049624731, −3.94782461609699127356225898855, −2.84318681768069752678120844869, −2.37637450964850415531118444975, −1.57781969362899684004733331654, 0, 1.57781969362899684004733331654, 2.37637450964850415531118444975, 2.84318681768069752678120844869, 3.94782461609699127356225898855, 4.40511957590511309406049624731, 5.69680529799440054433658676649, 5.99483726740740531161503807902, 6.76439742521779500910512964545, 7.66889360904242090685001518418

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