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  + 3·5-s + 7-s − 11-s + 2·13-s + 2·17-s − 3·19-s − 23-s + 4·25-s − 2·29-s − 5·31-s + 3·35-s + 7·37-s + 7·41-s + 8·43-s + 4·47-s + 49-s − 4·53-s − 3·55-s + 4·59-s + 4·61-s + 6·65-s + 2·67-s + 3·71-s − 77-s + 4·79-s + 2·83-s + 6·85-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.688·19-s − 0.208·23-s + 4/5·25-s − 0.371·29-s − 0.898·31-s + 0.507·35-s + 1.15·37-s + 1.09·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.549·53-s − 0.404·55-s + 0.520·59-s + 0.512·61-s + 0.744·65-s + 0.244·67-s + 0.356·71-s − 0.113·77-s + 0.450·79-s + 0.219·83-s + 0.650·85-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.880526897$
$L(\frac12)$  $\approx$  $2.880526897$
$L(\frac{3}{2})$   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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.031283237859020875819780729250, −7.44124513817703459179753597530, −6.46686870429689485960980611484, −5.87701928249365151076507487089, −5.42659410911837857479707052937, −4.49441729517652567002436442813, −3.66086872533175093927305837663, −2.52809508545776274555596918001, −1.95304382384510274888764059190, −0.920484220523452143920514394096, 0.920484220523452143920514394096, 1.95304382384510274888764059190, 2.52809508545776274555596918001, 3.66086872533175093927305837663, 4.49441729517652567002436442813, 5.42659410911837857479707052937, 5.87701928249365151076507487089, 6.46686870429689485960980611484, 7.44124513817703459179753597530, 8.031283237859020875819780729250

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