Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.76i·5-s i·7-s − 2.55·11-s − 2.32·13-s − 7.36i·17-s − 1.78i·19-s − 3.87·23-s − 2.64·25-s − 7.32i·29-s − 2.17i·31-s − 2.76·35-s − 1.24·37-s + 3.52i·41-s + 5.28i·43-s + 5.77·47-s + ⋯
L(s)  = 1  − 1.23i·5-s − 0.377i·7-s − 0.770·11-s − 0.645·13-s − 1.78i·17-s − 0.408i·19-s − 0.807·23-s − 0.529·25-s − 1.36i·29-s − 0.390i·31-s − 0.467·35-s − 0.205·37-s + 0.550i·41-s + 0.806i·43-s + 0.842·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.5563817764$
$L(\frac12)$  $\approx$  $0.5563817764$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 2.76iT - 5T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
13 \( 1 + 2.32T + 13T^{2} \)
17 \( 1 + 7.36iT - 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 + 7.32iT - 29T^{2} \)
31 \( 1 + 2.17iT - 31T^{2} \)
37 \( 1 + 1.24T + 37T^{2} \)
41 \( 1 - 3.52iT - 41T^{2} \)
43 \( 1 - 5.28iT - 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 - 7.29iT - 53T^{2} \)
59 \( 1 + 4.98T + 59T^{2} \)
61 \( 1 - 3.15T + 61T^{2} \)
67 \( 1 - 7.72iT - 67T^{2} \)
71 \( 1 - 7.98T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 2.10iT - 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 9.71iT - 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−7.61880198646234648937513180344, −7.16904011213309385161997447834, −6.06329829463602665860688743028, −5.36891311498259499294340638913, −4.65613709229020026510389871769, −4.29158599514658549193636094399, −2.97283020775687669178406960095, −2.26388292764439330377993565633, −0.976385572331423173464086848941, −0.15436328805167811404023252686, 1.72998757300110511806961548387, 2.44416066271502776411794347877, 3.31097480477827163573240852619, 3.93938611102401915592117431909, 5.06088341026894070337993849206, 5.73827534543079085424940716865, 6.43098079361527636258619447841, 7.08134484704226988647472904368, 7.77399456842273917743324761309, 8.401674786169622199224671904158

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