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  + 0.766·5-s + 7-s − 5.54·11-s + 6.31·13-s + 7.15·17-s − 6.15·19-s + 7.17·23-s − 4.41·25-s + 3.16·29-s − 0.163·31-s + 0.766·35-s + 5.31·37-s + 0.865·41-s − 8.31·43-s + 2.36·47-s + 49-s + 4·53-s − 4.24·55-s + 2.36·59-s − 13.5·61-s + 4.83·65-s + 7.05·67-s + 7.86·71-s − 1.98·73-s − 5.54·77-s + 5.24·79-s − 12.5·83-s + ⋯
L(s)  = 1  + 0.342·5-s + 0.377·7-s − 1.67·11-s + 1.75·13-s + 1.73·17-s − 1.41·19-s + 1.49·23-s − 0.882·25-s + 0.587·29-s − 0.0293·31-s + 0.129·35-s + 0.873·37-s + 0.135·41-s − 1.26·43-s + 0.345·47-s + 0.142·49-s + 0.549·53-s − 0.573·55-s + 0.308·59-s − 1.73·61-s + 0.599·65-s + 0.861·67-s + 0.932·71-s − 0.232·73-s − 0.632·77-s + 0.590·79-s − 1.37·83-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.290461132$
$L(\frac12)$  $\approx$  $2.290461132$
$L(\frac{3}{2})$   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 \( 1 - 0.766T + 5T^{2} \)
11 \( 1 + 5.54T + 11T^{2} \)
13 \( 1 - 6.31T + 13T^{2} \)
17 \( 1 - 7.15T + 17T^{2} \)
19 \( 1 + 6.15T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
31 \( 1 + 0.163T + 31T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 - 0.865T + 41T^{2} \)
43 \( 1 + 8.31T + 43T^{2} \)
47 \( 1 - 2.36T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 2.36T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 7.05T + 67T^{2} \)
71 \( 1 - 7.86T + 71T^{2} \)
73 \( 1 + 1.98T + 73T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 4.86T + 89T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−8.145519356406878934471737695441, −7.53738955541723694263613555291, −6.55476692426017732441988650991, −5.84108940292192155848020225484, −5.33663642764685398633458449749, −4.52029539024521554909237919194, −3.53468406791929501137330882824, −2.82455982816561083249617642606, −1.81830509337578447726443994116, −0.818436406134259403692567725934, 0.818436406134259403692567725934, 1.81830509337578447726443994116, 2.82455982816561083249617642606, 3.53468406791929501137330882824, 4.52029539024521554909237919194, 5.33663642764685398633458449749, 5.84108940292192155848020225484, 6.55476692426017732441988650991, 7.53738955541723694263613555291, 8.145519356406878934471737695441

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