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  =  $(2,\ 6048,\ (\ :1/2),\ 0.359 - 0.933i)$
$L(1)$  $\approx$  $0.7640916969$
$L(\frac12)$  $\approx$  $0.7640916969$
$L(\frac{3}{2})$   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

−8.087659890310824113437375965004, −7.79434213675650140803584084908, −6.76047101804973969573536146429, −6.04346611657383650056612398781, −5.46293343419416411762843562502, −4.42625108751074907785927222025, −3.79634774663639485330542497279, −3.08007876459752611852516958290, −2.12153050809292129741963040785, −0.73839132667634805591211698594, 0.27414636511902250888985365844, 1.71608066500172106406724688198, 2.53372303498945217355341310353, 3.81501610257247770517376195361, 4.16626164186867424774803988853, 4.81071975223443026328980501479, 5.87724719087858201989476311664, 6.68811750784413579081451618398, 7.31325070694768787387446761740, 7.84452241104305806483411267524

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