Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.717 - 0.696i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.206·5-s + i·7-s + 6.42i·11-s − 3.52i·13-s − 3.90i·17-s + 5.48·19-s − 0.880·23-s − 4.95·25-s + 3.20·29-s + 0.0631i·31-s − 0.206i·35-s − 9.91i·37-s + 5.94i·41-s + 1.13·43-s + 6.81·47-s + ⋯
L(s)  = 1  − 0.0925·5-s + 0.377i·7-s + 1.93i·11-s − 0.977i·13-s − 0.947i·17-s + 1.25·19-s − 0.183·23-s − 0.991·25-s + 0.595·29-s + 0.0113i·31-s − 0.0349i·35-s − 1.63i·37-s + 0.928i·41-s + 0.173·43-s + 0.993·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.717 - 0.696i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.717 - 0.696i)$
$L(1)$  $\approx$  $1.867038700$
$L(\frac12)$  $\approx$  $1.867038700$
$L(\frac{3}{2})$   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 \( 1 + 0.206T + 5T^{2} \)
11 \( 1 - 6.42iT - 11T^{2} \)
13 \( 1 + 3.52iT - 13T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 + 0.880T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 0.0631iT - 31T^{2} \)
37 \( 1 + 9.91iT - 37T^{2} \)
41 \( 1 - 5.94iT - 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 - 6.81T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 4.23iT - 59T^{2} \)
61 \( 1 - 12.3iT - 61T^{2} \)
67 \( 1 - 5.74T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 8.27T + 73T^{2} \)
79 \( 1 + 4.86iT - 79T^{2} \)
83 \( 1 - 7.51iT - 83T^{2} \)
89 \( 1 + 4.44iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−7.970094388010561142410619158362, −7.39628749463718607086794604657, −7.02514361040933503572285743783, −5.86414572906030333306967123158, −5.31776663544548508178698921211, −4.59555649008572400172179479518, −3.79125353312346454530051624674, −2.74545491073473301698770710871, −2.11228398966069030606924494599, −0.864639602710108507520842806846, 0.62523370935428357487292441887, 1.59454616068606894597631461948, 2.80105454197074600415833283861, 3.62822750941996135635566599471, 4.13333062206529207916964027300, 5.24109856312745433450202902382, 5.89212413476262577771314490426, 6.50169088202607671336204253501, 7.29526112723319477894503388347, 8.105920723857975935377367541280

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