Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24·5-s − 7-s − 5.74·11-s + 6.06·13-s − 2.56·17-s − 7.74·19-s − 1.67·23-s + 12.9·25-s − 5.24·29-s − 5.24·31-s + 4.24·35-s − 7.98·37-s − 2.24·41-s + 5.92·43-s − 6.62·47-s + 49-s − 2.85·53-s + 24.3·55-s − 7.48·59-s − 12.2·61-s − 25.7·65-s − 8.16·67-s − 2.32·71-s − 1.00·73-s + 5.74·77-s + 7.59·79-s + 13.9·83-s + ⋯
L(s)  = 1  − 1.89·5-s − 0.377·7-s − 1.73·11-s + 1.68·13-s − 0.621·17-s − 1.77·19-s − 0.350·23-s + 2.59·25-s − 0.973·29-s − 0.941·31-s + 0.716·35-s − 1.31·37-s − 0.349·41-s + 0.902·43-s − 0.966·47-s + 0.142·49-s − 0.392·53-s + 3.28·55-s − 0.973·59-s − 1.56·61-s − 3.18·65-s − 0.996·67-s − 0.275·71-s − 0.117·73-s + 0.654·77-s + 0.855·79-s + 1.53·83-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  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2395192683$
$L(\frac12)$  $\approx$  $0.2395192683$
$L(\frac{3}{2})$   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 \( 1 + 4.24T + 5T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + 7.74T + 19T^{2} \)
23 \( 1 + 1.67T + 23T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 + 5.24T + 31T^{2} \)
37 \( 1 + 7.98T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 5.92T + 43T^{2} \)
47 \( 1 + 6.62T + 47T^{2} \)
53 \( 1 + 2.85T + 53T^{2} \)
59 \( 1 + 7.48T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 8.16T + 67T^{2} \)
71 \( 1 + 2.32T + 71T^{2} \)
73 \( 1 + 1.00T + 73T^{2} \)
79 \( 1 - 7.59T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 3.61T + 97T^{2} \)
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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

−7.953223420576754682137860633558, −7.62581425553914766645189766187, −6.70808635411030201851067522555, −6.05466961891714885537460490692, −5.05617150893698857875816532821, −4.28922706954406643639333691619, −3.66607813458766856062279296097, −3.04987047852037760522427943738, −1.87128340978240931319335659275, −0.24363135912001096553816149777, 0.24363135912001096553816149777, 1.87128340978240931319335659275, 3.04987047852037760522427943738, 3.66607813458766856062279296097, 4.28922706954406643639333691619, 5.05617150893698857875816532821, 6.05466961891714885537460490692, 6.70808635411030201851067522555, 7.62581425553914766645189766187, 7.953223420576754682137860633558

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