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  − 3.82·5-s − 7-s + 0.828·11-s − 2.82·13-s + 5.82·17-s − 1.17·19-s + 4·23-s + 9.65·25-s − 2.82·29-s + 1.17·31-s + 3.82·35-s − 8.65·37-s − 1.82·41-s − 2.17·43-s + 2.65·47-s + 49-s − 2·53-s − 3.17·55-s + 10.6·59-s + 8.82·61-s + 10.8·65-s − 4·67-s + 9.65·71-s + 5.65·73-s − 0.828·77-s − 3.82·79-s + 16.6·83-s + ⋯
L(s)  = 1  − 1.71·5-s − 0.377·7-s + 0.249·11-s − 0.784·13-s + 1.41·17-s − 0.268·19-s + 0.834·23-s + 1.93·25-s − 0.525·29-s + 0.210·31-s + 0.647·35-s − 1.42·37-s − 0.285·41-s − 0.331·43-s + 0.387·47-s + 0.142·49-s − 0.274·53-s − 0.427·55-s + 1.38·59-s + 1.13·61-s + 1.34·65-s − 0.488·67-s + 1.14·71-s + 0.662·73-s − 0.0944·77-s − 0.430·79-s + 1.82·83-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 \( 1 + 3.82T + 5T^{2} \)
11 \( 1 - 0.828T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 5.82T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 8.65T + 37T^{2} \)
41 \( 1 + 1.82T + 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 - 2.65T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 9.65T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 + 3.82T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−7.67496317354485611205330557650, −7.14691619505317162786814754751, −6.54880183920523917831092602101, −5.35875498371719866755475551393, −4.85635420594487085711692111717, −3.75047525423384132373038840596, −3.53650958946091510643256105470, −2.51132324812274949222030472644, −1.05942745434630657319277593251, 0, 1.05942745434630657319277593251, 2.51132324812274949222030472644, 3.53650958946091510643256105470, 3.75047525423384132373038840596, 4.85635420594487085711692111717, 5.35875498371719866755475551393, 6.54880183920523917831092602101, 7.14691619505317162786814754751, 7.67496317354485611205330557650

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