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 + 2·11-s + 17-s + 4·19-s − 4·23-s − 4·25-s − 8·29-s − 4·31-s − 35-s − 7·37-s − 5·41-s − 43-s − 9·47-s + 49-s + 2·53-s + 2·55-s + 3·59-s − 2·61-s + 4·67-s + 12·71-s − 16·73-s − 2·77-s + 15·79-s − 3·83-s + 85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 4/5·25-s − 1.48·29-s − 0.718·31-s − 0.169·35-s − 1.15·37-s − 0.780·41-s − 0.152·43-s − 1.31·47-s + 1/7·49-s + 0.274·53-s + 0.269·55-s + 0.390·59-s − 0.256·61-s + 0.488·67-s + 1.42·71-s − 1.87·73-s − 0.227·77-s + 1.68·79-s − 0.329·83-s + 0.108·85-s + 0.635·89-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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.65004329832495213018292943327, −7.02149239656734196114444400280, −6.25575024261827051989994213029, −5.61510414361084546603037642384, −4.98291454268829879515680501784, −3.78536600467766840643531586830, −3.44253040392140435550826615231, −2.18628651419004802539130214164, −1.45766279791695892903734607199, 0, 1.45766279791695892903734607199, 2.18628651419004802539130214164, 3.44253040392140435550826615231, 3.78536600467766840643531586830, 4.98291454268829879515680501784, 5.61510414361084546603037642384, 6.25575024261827051989994213029, 7.02149239656734196114444400280, 7.65004329832495213018292943327

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