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  − 4.29i·5-s i·7-s − 4.77·11-s − 0.0836·13-s − 2.60i·17-s − 4.85i·19-s + 4.78·23-s − 13.4·25-s + 0.263i·29-s − 0.304i·31-s − 4.29·35-s + 9.56·37-s − 10.5i·41-s + 4.24i·43-s − 9.85·47-s + ⋯
L(s)  = 1  − 1.92i·5-s − 0.377i·7-s − 1.43·11-s − 0.0231·13-s − 0.632i·17-s − 1.11i·19-s + 0.996·23-s − 2.69·25-s + 0.0488i·29-s − 0.0546i·31-s − 0.726·35-s + 1.57·37-s − 1.65i·41-s + 0.647i·43-s − 1.43·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.8549924311$
$L(\frac12)$  $\approx$  $0.8549924311$
$L(\frac{3}{2})$   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 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 + 0.0836T + 13T^{2} \)
17 \( 1 + 2.60iT - 17T^{2} \)
19 \( 1 + 4.85iT - 19T^{2} \)
23 \( 1 - 4.78T + 23T^{2} \)
29 \( 1 - 0.263iT - 29T^{2} \)
31 \( 1 + 0.304iT - 31T^{2} \)
37 \( 1 - 9.56T + 37T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + 9.85T + 47T^{2} \)
53 \( 1 + 11.9iT - 53T^{2} \)
59 \( 1 - 5.50T + 59T^{2} \)
61 \( 1 + 8.62T + 61T^{2} \)
67 \( 1 + 1.80iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 4.88T + 73T^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + 7.85iT - 89T^{2} \)
97 \( 1 + 1.29T + 97T^{2} \)
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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

−7.72523387124255530758134327087, −7.12972727522072825815405020839, −6.03620828287775516642692144026, −5.09430248949306407229840007823, −4.98749897418931503585900724647, −4.21176219993687116799335647940, −3.07442816261977578157041197100, −2.12874180197534926444491311027, −0.961752697557961792999523556966, −0.24310786347430728353857114589, 1.70250136152848979358903952546, 2.78280029329075592525691266367, 2.96957178842509979737541940422, 3.98596594091835624998261590244, 4.99592114848699685556704752222, 5.99508031646012975667187875609, 6.21974740075884577652106537651, 7.18903248212038713479341764241, 7.76896572091914993426998675080, 8.196818610240187970186856889753

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