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  + 3.95·5-s − 7-s − 0.848·11-s − 5.85·13-s + 2.75·17-s − 7.66·19-s + 2.70·23-s + 10.6·25-s − 5.85·29-s − 5.80·31-s − 3.95·35-s − 3.35·37-s + 1.90·41-s − 2.35·43-s + 3.55·47-s + 49-s + 10.2·53-s − 3.35·55-s + 5.77·59-s − 5.50·61-s − 23.1·65-s − 11.4·67-s − 9.45·71-s − 7.97·73-s + 0.848·77-s − 10.0·79-s − 9.85·83-s + ⋯
L(s)  = 1  + 1.77·5-s − 0.377·7-s − 0.255·11-s − 1.62·13-s + 0.666·17-s − 1.75·19-s + 0.564·23-s + 2.13·25-s − 1.08·29-s − 1.04·31-s − 0.669·35-s − 0.552·37-s + 0.296·41-s − 0.359·43-s + 0.518·47-s + 0.142·49-s + 1.40·53-s − 0.452·55-s + 0.751·59-s − 0.704·61-s − 2.87·65-s − 1.40·67-s − 1.12·71-s − 0.933·73-s + 0.0967·77-s − 1.13·79-s − 1.08·83-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 \( 1 - 3.95T + 5T^{2} \)
11 \( 1 + 0.848T + 11T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 + 7.66T + 19T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + 3.35T + 37T^{2} \)
41 \( 1 - 1.90T + 41T^{2} \)
43 \( 1 + 2.35T + 43T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 - 5.77T + 59T^{2} \)
61 \( 1 + 5.50T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 9.45T + 71T^{2} \)
73 \( 1 + 7.97T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 9.85T + 83T^{2} \)
89 \( 1 + 1.81T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−7.47484946997623901858437482439, −7.04245007100042627909162039913, −6.17502573937938522782221174557, −5.60200748333670418485119132532, −5.05980781865816419816062421611, −4.13216308107884487842197387466, −2.91342307248750192133693783016, −2.30681072293080591246846384673, −1.58397742919684242623867525758, 0, 1.58397742919684242623867525758, 2.30681072293080591246846384673, 2.91342307248750192133693783016, 4.13216308107884487842197387466, 5.05980781865816419816062421611, 5.60200748333670418485119132532, 6.17502573937938522782221174557, 7.04245007100042627909162039913, 7.47484946997623901858437482439

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