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  − 1.96·5-s + 7-s − 3.38·11-s + 6.48·13-s + 5.06·17-s − 4.13·19-s − 7.09·23-s − 1.13·25-s − 6.48·29-s + 6.34·31-s − 1.96·35-s − 6.65·37-s + 8.44·41-s + 5.65·43-s − 3.71·47-s + 49-s − 1.17·53-s + 6.65·55-s − 10.5·59-s + 10.1·61-s − 12.7·65-s + 0.216·67-s + 8.16·71-s − 12.3·73-s − 3.38·77-s − 11.3·79-s + 2.48·83-s + ⋯
L(s)  = 1  − 0.879·5-s + 0.377·7-s − 1.01·11-s + 1.79·13-s + 1.22·17-s − 0.948·19-s − 1.48·23-s − 0.226·25-s − 1.20·29-s + 1.14·31-s − 0.332·35-s − 1.09·37-s + 1.31·41-s + 0.861·43-s − 0.542·47-s + 0.142·49-s − 0.160·53-s + 0.896·55-s − 1.37·59-s + 1.29·61-s − 1.58·65-s + 0.0263·67-s + 0.968·71-s − 1.44·73-s − 0.385·77-s − 1.27·79-s + 0.272·83-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  =  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 \( 1 + 1.96T + 5T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
23 \( 1 + 7.09T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 + 6.65T + 37T^{2} \)
41 \( 1 - 8.44T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + 3.71T + 47T^{2} \)
53 \( 1 + 1.17T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 0.216T + 67T^{2} \)
71 \( 1 - 8.16T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 7.86T + 97T^{2} \)
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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

−7.892302724430374292362758071941, −7.23549633618844797838946055653, −6.00485720056768267521795581375, −5.84022284715977683623071890315, −4.70821623877111662651496583957, −3.92907537113808629967662038523, −3.45338333184067653435572420290, −2.31559528601382538606644926080, −1.26754755950454492710176783200, 0, 1.26754755950454492710176783200, 2.31559528601382538606644926080, 3.45338333184067653435572420290, 3.92907537113808629967662038523, 4.70821623877111662651496583957, 5.84022284715977683623071890315, 6.00485720056768267521795581375, 7.23549633618844797838946055653, 7.892302724430374292362758071941

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