Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.880 + 0.474i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.66i·5-s − 7-s + 4.45i·11-s + 1.51i·13-s − 3.45·17-s − 2.29i·19-s + 8.76·23-s − 8.44·25-s + 1.62i·29-s + 9.26·31-s + 3.66i·35-s − 5.69i·37-s − 6.86·41-s − 0.880i·43-s − 10.5·47-s + ⋯
L(s)  = 1  − 1.63i·5-s − 0.377·7-s + 1.34i·11-s + 0.420i·13-s − 0.837·17-s − 0.527i·19-s + 1.82·23-s − 1.68·25-s + 0.300i·29-s + 1.66·31-s + 0.619i·35-s − 0.936i·37-s − 1.07·41-s − 0.134i·43-s − 1.54·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.880 + 0.474i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.880 + 0.474i)$
$L(1)$  $\approx$  $1.003831294$
$L(\frac12)$  $\approx$  $1.003831294$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 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 \( 1 + 3.66iT - 5T^{2} \)
11 \( 1 - 4.45iT - 11T^{2} \)
13 \( 1 - 1.51iT - 13T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 - 8.76T + 23T^{2} \)
29 \( 1 - 1.62iT - 29T^{2} \)
31 \( 1 - 9.26T + 31T^{2} \)
37 \( 1 + 5.69iT - 37T^{2} \)
41 \( 1 + 6.86T + 41T^{2} \)
43 \( 1 + 0.880iT - 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + 2.99iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 + 4.57T + 79T^{2} \)
83 \( 1 - 2.60iT - 83T^{2} \)
89 \( 1 + 2.14T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.911395547063765761832474269409, −6.87786874117154897002481374243, −6.61169064720030160906777200995, −5.33873016758857613662967626568, −4.76502909976467373765067201725, −4.48132984457218947732232457919, −3.35234132266219115910377936275, −2.20526967226756551406784048335, −1.39301492693064228055908485760, −0.27056365220982261768726980441, 1.17921749321237595744301752593, 2.79810351582421021456882381820, 2.87532843171269078729704643972, 3.72441610103068097077438774717, 4.75785417984127021433987718996, 5.77007359878026600564853256130, 6.38490018289435968097613095500, 6.79929712708923047601110714743, 7.56320118879178444157598264447, 8.379694891712599598844130342383

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