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  + 7-s − 4·11-s + 13-s − 5·17-s − 2·19-s + 3·23-s − 5·25-s + 29-s + 9·31-s + 6·37-s + 2·41-s − 43-s − 2·47-s + 49-s − 9·53-s − 3·59-s − 2·61-s + 11·67-s + 5·71-s − 2·73-s − 4·77-s + 14·79-s + 12·83-s + 15·89-s + 91-s + 12·97-s + 101-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.20·11-s + 0.277·13-s − 1.21·17-s − 0.458·19-s + 0.625·23-s − 25-s + 0.185·29-s + 1.61·31-s + 0.986·37-s + 0.312·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s − 1.23·53-s − 0.390·59-s − 0.256·61-s + 1.34·67-s + 0.593·71-s − 0.234·73-s − 0.455·77-s + 1.57·79-s + 1.31·83-s + 1.58·89-s + 0.104·91-s + 1.21·97-s + 0.0995·101-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  :  $\chi_{6048} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.623450144$
$L(\frac12)$  $\approx$  $1.623450144$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−17.49178259246799, −16.94646132853188, −16.07875252265711, −15.56094029266010, −15.25682109608141, −14.43796296660080, −13.69571644275680, −13.26802031157818, −12.73271998192485, −11.92423446016359, −11.21678344239595, −10.81527261094158, −10.11143018936696, −9.413083435244391, −8.651001284191721, −8.014024272160446, −7.569251646397940, −6.491585877595249, −6.113890382976710, −4.970452574990600, −4.664516666897487, −3.647196954970348, −2.642971165562297, −2.008124234630946, −0.6502320973807721, 0.6502320973807721, 2.008124234630946, 2.642971165562297, 3.647196954970348, 4.664516666897487, 4.970452574990600, 6.113890382976710, 6.491585877595249, 7.569251646397940, 8.014024272160446, 8.651001284191721, 9.413083435244391, 10.11143018936696, 10.81527261094158, 11.21678344239595, 11.92423446016359, 12.73271998192485, 13.26802031157818, 13.69571644275680, 14.43796296660080, 15.25682109608141, 15.56094029266010, 16.07875252265711, 16.94646132853188, 17.49178259246799

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