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  − 5-s − 7-s − 3·11-s + 2·13-s + 2·17-s − 5·19-s − 3·23-s − 4·25-s + 6·29-s − 3·31-s + 35-s − 9·37-s + 3·41-s + 8·43-s − 4·47-s + 49-s + 12·53-s + 3·55-s + 12·59-s + 4·61-s − 2·65-s − 2·67-s + 71-s + 3·77-s − 4·79-s + 6·83-s − 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.554·13-s + 0.485·17-s − 1.14·19-s − 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.538·31-s + 0.169·35-s − 1.47·37-s + 0.468·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.64·53-s + 0.404·55-s + 1.56·59-s + 0.512·61-s − 0.248·65-s − 0.244·67-s + 0.118·71-s + 0.341·77-s − 0.450·79-s + 0.658·83-s − 0.216·85-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.189865185$
$L(\frac12)$  $\approx$  $1.189865185$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.220858251567369747720351025593, −7.36826673824308526454828931511, −6.72294780352544673166965391195, −5.87290623867614865610012498158, −5.33417309427618127516073741487, −4.26110230817488576814376855650, −3.74621421231830989388764114361, −2.79109241798921227695560595507, −1.93834041911221566651566608065, −0.55307393407904666798208817107, 0.55307393407904666798208817107, 1.93834041911221566651566608065, 2.79109241798921227695560595507, 3.74621421231830989388764114361, 4.26110230817488576814376855650, 5.33417309427618127516073741487, 5.87290623867614865610012498158, 6.72294780352544673166965391195, 7.36826673824308526454828931511, 8.220858251567369747720351025593

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