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  + 5-s + 7-s + 2·11-s + 4·13-s − 3·17-s − 4·19-s − 4·25-s − 8·29-s − 8·31-s + 35-s − 11·37-s + 7·41-s − 11·43-s − 11·47-s + 49-s + 2·53-s + 2·55-s − 11·59-s − 6·61-s + 4·65-s + 4·67-s + 4·71-s + 12·73-s + 2·77-s + 79-s + 3·83-s − 3·85-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.603·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 4/5·25-s − 1.48·29-s − 1.43·31-s + 0.169·35-s − 1.80·37-s + 1.09·41-s − 1.67·43-s − 1.60·47-s + 1/7·49-s + 0.274·53-s + 0.269·55-s − 1.43·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.474·71-s + 1.40·73-s + 0.227·77-s + 0.112·79-s + 0.329·83-s − 0.325·85-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  =  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 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−7.78003256011758829135721926648, −6.88800429358339012062396573927, −6.31627192154263596582854816570, −5.64674247274923299113665412668, −4.86884650915969867870030183306, −3.91948186621161182996928468018, −3.42919244013805701404858993129, −1.97366780564397302648658117358, −1.64113782166109996907301654566, 0, 1.64113782166109996907301654566, 1.97366780564397302648658117358, 3.42919244013805701404858993129, 3.91948186621161182996928468018, 4.86884650915969867870030183306, 5.64674247274923299113665412668, 6.31627192154263596582854816570, 6.88800429358339012062396573927, 7.78003256011758829135721926648

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