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  − 1.56·5-s − 7-s − 3.12·11-s − 6.56·13-s − 4.12·17-s − 5.12·19-s − 3.68·23-s − 2.56·25-s + 5.43·29-s − 2.56·31-s + 1.56·35-s + 3.56·37-s − 11.5·41-s + 9.24·43-s − 3.56·47-s + 49-s + 5.68·53-s + 4.87·55-s + 0.123·59-s + 8.24·61-s + 10.2·65-s + 15.6·67-s − 2.56·71-s + 1.12·73-s + 3.12·77-s + 4.43·79-s − 1.56·83-s + ⋯
L(s)  = 1  − 0.698·5-s − 0.377·7-s − 0.941·11-s − 1.81·13-s − 0.999·17-s − 1.17·19-s − 0.768·23-s − 0.512·25-s + 1.00·29-s − 0.460·31-s + 0.263·35-s + 0.585·37-s − 1.80·41-s + 1.41·43-s − 0.519·47-s + 0.142·49-s + 0.780·53-s + 0.657·55-s + 0.0160·59-s + 1.05·61-s + 1.27·65-s + 1.91·67-s − 0.304·71-s + 0.131·73-s + 0.355·77-s + 0.499·79-s − 0.171·83-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4012754563$
$L(\frac12)$  $\approx$  $0.4012754563$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 1.56T + 5T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 + 6.56T + 13T^{2} \)
17 \( 1 + 4.12T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 - 3.56T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + 3.56T + 47T^{2} \)
53 \( 1 - 5.68T + 53T^{2} \)
59 \( 1 - 0.123T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 - 4.43T + 79T^{2} \)
83 \( 1 + 1.56T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−8.185037566124457454187965356609, −7.28335805555971787475607674676, −6.86179672765516655668586464395, −5.95884525200823944220628810270, −5.06619148999116622694740139628, −4.46123939242294234878767897192, −3.73985436468492827761787321342, −2.58925734690730868482804582221, −2.17154085802695561029573545373, −0.30485207844741623894514425027, 0.30485207844741623894514425027, 2.17154085802695561029573545373, 2.58925734690730868482804582221, 3.73985436468492827761787321342, 4.46123939242294234878767897192, 5.06619148999116622694740139628, 5.95884525200823944220628810270, 6.86179672765516655668586464395, 7.28335805555971787475607674676, 8.185037566124457454187965356609

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