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 + 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  =  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 \( 1 - 3 T + p T^{2} \)
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

−7.63073433009683807046376086845, −6.93814463682815502703059146648, −6.23125124944916613687003283240, −5.80030797985659329670974132752, −4.78318803766113629917092646138, −3.96712457702569468317619745446, −3.34306190957315203820289153776, −2.18278176057228505111518464734, −1.46705543408855278037660137170, 0, 1.46705543408855278037660137170, 2.18278176057228505111518464734, 3.34306190957315203820289153776, 3.96712457702569468317619745446, 4.78318803766113629917092646138, 5.80030797985659329670974132752, 6.23125124944916613687003283240, 6.93814463682815502703059146648, 7.63073433009683807046376086845

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