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  + 2.73·5-s − 7-s − 2.24·11-s − 0.458·13-s + 7.44·17-s − 4.24·19-s − 4.70·23-s + 2.50·25-s + 1.73·29-s + 1.73·31-s − 2.73·35-s + 2.49·37-s + 4.73·41-s + 1.96·43-s + 9.90·47-s + 49-s − 5.42·53-s − 6.15·55-s + 6.47·59-s + 5.23·61-s − 1.25·65-s + 2.77·67-s + 0.703·71-s − 7.96·73-s + 2.24·77-s + 6.66·79-s + 3.50·83-s + ⋯
L(s)  = 1  + 1.22·5-s − 0.377·7-s − 0.676·11-s − 0.127·13-s + 1.80·17-s − 0.973·19-s − 0.980·23-s + 0.501·25-s + 0.323·29-s + 0.312·31-s − 0.463·35-s + 0.410·37-s + 0.740·41-s + 0.299·43-s + 1.44·47-s + 0.142·49-s − 0.744·53-s − 0.829·55-s + 0.843·59-s + 0.670·61-s − 0.155·65-s + 0.339·67-s + 0.0835·71-s − 0.932·73-s + 0.255·77-s + 0.750·79-s + 0.384·83-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  =  $(2,\ 6048,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.347952391$
$L(\frac12)$  $\approx$  $2.347952391$
$L(\frac{3}{2})$   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 \( 1 - 2.73T + 5T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 0.458T + 13T^{2} \)
17 \( 1 - 7.44T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 1.73T + 31T^{2} \)
37 \( 1 - 2.49T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 1.96T + 43T^{2} \)
47 \( 1 - 9.90T + 47T^{2} \)
53 \( 1 + 5.42T + 53T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 - 2.77T + 67T^{2} \)
71 \( 1 - 0.703T + 71T^{2} \)
73 \( 1 + 7.96T + 73T^{2} \)
79 \( 1 - 6.66T + 79T^{2} \)
83 \( 1 - 3.50T + 83T^{2} \)
89 \( 1 - 1.78T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.004337099602067058845335336822, −7.45343575076716841674316065442, −6.45842404913301035392523976153, −5.87976555491823846728735587947, −5.45772182846955295634266443687, −4.50532685950744910243341219794, −3.55907727146283424361916910823, −2.62541738646179927831798451508, −1.99003652134668059131515679829, −0.802543593110700039256812044837, 0.802543593110700039256812044837, 1.99003652134668059131515679829, 2.62541738646179927831798451508, 3.55907727146283424361916910823, 4.50532685950744910243341219794, 5.45772182846955295634266443687, 5.87976555491823846728735587947, 6.45842404913301035392523976153, 7.45343575076716841674316065442, 8.004337099602067058845335336822

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