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  − 0.414·5-s − 7-s + 0.414·11-s − 2.82·13-s − 2.82·17-s + 1.82·19-s + 3.24·23-s − 4.82·25-s + 2.82·29-s + 6.65·31-s + 0.414·35-s + 5.82·37-s + 12.0·41-s + 0.828·43-s − 11.6·47-s + 49-s − 4·53-s − 0.171·55-s − 7.65·59-s + 11.3·61-s + 1.17·65-s − 12.4·67-s − 8.41·71-s − 8.82·73-s − 0.414·77-s − 0.828·79-s − 5.17·83-s + ⋯
L(s)  = 1  − 0.185·5-s − 0.377·7-s + 0.124·11-s − 0.784·13-s − 0.685·17-s + 0.419·19-s + 0.676·23-s − 0.965·25-s + 0.525·29-s + 1.19·31-s + 0.0700·35-s + 0.958·37-s + 1.88·41-s + 0.126·43-s − 1.70·47-s + 0.142·49-s − 0.549·53-s − 0.0231·55-s − 0.996·59-s + 1.44·61-s + 0.145·65-s − 1.52·67-s − 0.998·71-s − 1.03·73-s − 0.0472·77-s − 0.0932·79-s − 0.567·83-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 \( 1 + 0.414T + 5T^{2} \)
11 \( 1 - 0.414T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 - 5.82T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 0.828T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 7.65T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 + 8.82T + 73T^{2} \)
79 \( 1 + 0.828T + 79T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 + 7.24T + 89T^{2} \)
97 \( 1 - 1.65T + 97T^{2} \)
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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

−7.72909545769485688276484389422, −7.02008603442647904049580866077, −6.33850075272187957624060300598, −5.65089517907392505334987251137, −4.65810617440404148135119447060, −4.21400291355676059759753344580, −3.06859827210896982320436243239, −2.50623516113942303769594646027, −1.25758845434102922817243520612, 0, 1.25758845434102922817243520612, 2.50623516113942303769594646027, 3.06859827210896982320436243239, 4.21400291355676059759753344580, 4.65810617440404148135119447060, 5.65089517907392505334987251137, 6.33850075272187957624060300598, 7.02008603442647904049580866077, 7.72909545769485688276484389422

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