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  − 2.56·5-s + 7-s + 5.12·11-s − 2.43·13-s − 4.12·17-s − 3.12·19-s + 8.68·23-s + 1.56·25-s − 9.56·29-s − 1.56·31-s − 2.56·35-s − 0.561·37-s + 7.43·41-s + 7.24·43-s + 0.561·47-s + 49-s + 6.68·53-s − 13.1·55-s − 8.12·59-s − 8.24·61-s + 6.24·65-s − 3.31·67-s + 1.56·71-s − 7.12·73-s + 5.12·77-s − 8.56·79-s + 2.56·83-s + ⋯
L(s)  = 1  − 1.14·5-s + 0.377·7-s + 1.54·11-s − 0.676·13-s − 0.999·17-s − 0.716·19-s + 1.81·23-s + 0.312·25-s − 1.77·29-s − 0.280·31-s − 0.432·35-s − 0.0923·37-s + 1.16·41-s + 1.10·43-s + 0.0819·47-s + 0.142·49-s + 0.918·53-s − 1.76·55-s − 1.05·59-s − 1.05·61-s + 0.774·65-s − 0.405·67-s + 0.185·71-s − 0.833·73-s + 0.583·77-s − 0.963·79-s + 0.281·83-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 \( 1 + 2.56T + 5T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 + 2.43T + 13T^{2} \)
17 \( 1 + 4.12T + 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 - 8.68T + 23T^{2} \)
29 \( 1 + 9.56T + 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + 0.561T + 37T^{2} \)
41 \( 1 - 7.43T + 41T^{2} \)
43 \( 1 - 7.24T + 43T^{2} \)
47 \( 1 - 0.561T + 47T^{2} \)
53 \( 1 - 6.68T + 53T^{2} \)
59 \( 1 + 8.12T + 59T^{2} \)
61 \( 1 + 8.24T + 61T^{2} \)
67 \( 1 + 3.31T + 67T^{2} \)
71 \( 1 - 1.56T + 71T^{2} \)
73 \( 1 + 7.12T + 73T^{2} \)
79 \( 1 + 8.56T + 79T^{2} \)
83 \( 1 - 2.56T + 83T^{2} \)
89 \( 1 - 1.31T + 89T^{2} \)
97 \( 1 + 1.75T + 97T^{2} \)
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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

−7.46041479001014593241478061909, −7.26444506304166993370996765374, −6.43283918930447401889335975553, −5.56815877437785711257551015919, −4.48185979240217685196597872859, −4.22009130588586958134212661837, −3.37997415252456926925137901879, −2.32016830210195000109132112181, −1.24966116164055530999661264810, 0, 1.24966116164055530999661264810, 2.32016830210195000109132112181, 3.37997415252456926925137901879, 4.22009130588586958134212661837, 4.48185979240217685196597872859, 5.56815877437785711257551015919, 6.43283918930447401889335975553, 7.26444506304166993370996765374, 7.46041479001014593241478061909

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