Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.82·5-s − 7-s − 4.82·11-s + 2.82·13-s + 0.171·17-s − 6.82·19-s + 4·23-s − 1.65·25-s + 2.82·29-s + 6.82·31-s − 1.82·35-s + 2.65·37-s + 3.82·41-s − 7.82·43-s − 8.65·47-s + 49-s − 2·53-s − 8.82·55-s − 0.656·59-s + 3.17·61-s + 5.17·65-s − 4·67-s − 1.65·71-s − 5.65·73-s + 4.82·77-s + 1.82·79-s + 5.34·83-s + ⋯
L(s)  = 1  + 0.817·5-s − 0.377·7-s − 1.45·11-s + 0.784·13-s + 0.0416·17-s − 1.56·19-s + 0.834·23-s − 0.331·25-s + 0.525·29-s + 1.22·31-s − 0.309·35-s + 0.436·37-s + 0.597·41-s − 1.19·43-s − 1.26·47-s + 0.142·49-s − 0.274·53-s − 1.19·55-s − 0.0855·59-s + 0.406·61-s + 0.641·65-s − 0.488·67-s − 0.196·71-s − 0.662·73-s + 0.550·77-s + 0.205·79-s + 0.586·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 1.82T + 5T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 0.171T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 - 2.65T + 37T^{2} \)
41 \( 1 - 3.82T + 41T^{2} \)
43 \( 1 + 7.82T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 0.656T + 59T^{2} \)
61 \( 1 - 3.17T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 + 5.65T + 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 - 5.34T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−7.940251506125181964542625213198, −6.77298811774642900871713088120, −6.34995240345156980337986739286, −5.62417022239354461906383125881, −4.91841505741513666894215540823, −4.10583470214326168171154076130, −2.99228183304057857889543772363, −2.44097987761418239959035528206, −1.39680333501336359637658968782, 0, 1.39680333501336359637658968782, 2.44097987761418239959035528206, 2.99228183304057857889543772363, 4.10583470214326168171154076130, 4.91841505741513666894215540823, 5.62417022239354461906383125881, 6.34995240345156980337986739286, 6.77298811774642900871713088120, 7.940251506125181964542625213198

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