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  − 2.58·5-s + 7-s + 4.04·11-s − 0.190·13-s + 6.82·17-s − 1.34·19-s − 0.240·23-s + 1.65·25-s − 0.190·29-s + 2.46·31-s − 2.58·35-s + 10.4·37-s + 2.77·41-s − 11.4·43-s − 4.29·47-s + 49-s − 9.26·53-s − 10.4·55-s + 12.9·59-s − 13.6·61-s + 0.490·65-s − 0.871·67-s + 11.0·71-s + 12.5·73-s + 4.04·77-s − 15.0·79-s + 4.19·83-s + ⋯
L(s)  = 1  − 1.15·5-s + 0.377·7-s + 1.22·11-s − 0.0527·13-s + 1.65·17-s − 0.307·19-s − 0.0500·23-s + 0.331·25-s − 0.0353·29-s + 0.443·31-s − 0.436·35-s + 1.71·37-s + 0.432·41-s − 1.74·43-s − 0.625·47-s + 0.142·49-s − 1.27·53-s − 1.40·55-s + 1.68·59-s − 1.74·61-s + 0.0608·65-s − 0.106·67-s + 1.31·71-s + 1.46·73-s + 0.461·77-s − 1.69·79-s + 0.459·83-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.806641258$
$L(\frac12)$  $\approx$  $1.806641258$
$L(\frac{3}{2})$   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 \( 1 + 2.58T + 5T^{2} \)
11 \( 1 - 4.04T + 11T^{2} \)
13 \( 1 + 0.190T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 1.34T + 19T^{2} \)
23 \( 1 + 0.240T + 23T^{2} \)
29 \( 1 + 0.190T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 2.77T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + 4.29T + 47T^{2} \)
53 \( 1 + 9.26T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 0.871T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 4.19T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−8.035923231766130191463837597323, −7.53828348766238353176402733898, −6.69376299240256899032147448857, −6.03687818618251709091688457049, −5.08099565591386451220694979339, −4.32539563792993796321515971365, −3.70060737761899590318465919726, −3.01306509434009332523956426195, −1.67594470274124106960422845604, −0.74367617405895369296108230771, 0.74367617405895369296108230771, 1.67594470274124106960422845604, 3.01306509434009332523956426195, 3.70060737761899590318465919726, 4.32539563792993796321515971365, 5.08099565591386451220694979339, 6.03687818618251709091688457049, 6.69376299240256899032147448857, 7.53828348766238353176402733898, 8.035923231766130191463837597323

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