Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.359 + 0.933i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.29·5-s i·7-s − 4.02i·11-s + 1.82i·13-s + 0.430i·17-s − 5.01·19-s + 3.49·23-s + 0.274·25-s + 2.16·29-s − 2.10i·31-s − 2.29i·35-s + 2.19i·37-s + 4.35i·41-s + 12.0·43-s − 1.72·47-s + ⋯
L(s)  = 1  + 1.02·5-s − 0.377i·7-s − 1.21i·11-s + 0.506i·13-s + 0.104i·17-s − 1.14·19-s + 0.728·23-s + 0.0548·25-s + 0.401·29-s − 0.378i·31-s − 0.388i·35-s + 0.361i·37-s + 0.679i·41-s + 1.83·43-s − 0.251·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.359 + 0.933i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.359 + 0.933i)$
$L(1)$  $\approx$  $2.210684709$
$L(\frac12)$  $\approx$  $2.210684709$
$L(\frac{3}{2})$   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 \( 1 - 2.29T + 5T^{2} \)
11 \( 1 + 4.02iT - 11T^{2} \)
13 \( 1 - 1.82iT - 13T^{2} \)
17 \( 1 - 0.430iT - 17T^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 + 2.10iT - 31T^{2} \)
37 \( 1 - 2.19iT - 37T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 + 8.44iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 + 15.2iT - 79T^{2} \)
83 \( 1 + 13.4iT - 83T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−8.003634158013348498587980289262, −7.15923014764845413631481164142, −6.23108631316597668148255841386, −6.09190781175015727271974683909, −5.10141998167936091623921234095, −4.32168206281954867137145645023, −3.44608539550108009979942687632, −2.53803171059597854763468702386, −1.69471227447157148022727772988, −0.58484955435015787041544301465, 1.11274547111430939458253153087, 2.24124328736547502653919069310, 2.57509222336722035136252982175, 3.90508238686934805540216517334, 4.64307905650750256882308046233, 5.49747793842201174750875438849, 5.93177582648404346705724466295, 6.86998649735055059029633889530, 7.32183976789085208321285954549, 8.327410539560169470770881820812

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