Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.965 + 0.258i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.38·5-s i·7-s + 1.65i·11-s + 0.253i·13-s − 2i·17-s − 7.03·19-s + 2.82·23-s + 0.692·25-s + 1.26·29-s − 0.746i·31-s + 2.38i·35-s + 6.94i·37-s + 5.55i·41-s − 8.67·43-s + 10.9·47-s + ⋯
L(s)  = 1  − 1.06·5-s − 0.377i·7-s + 0.498i·11-s + 0.0702i·13-s − 0.485i·17-s − 1.61·19-s + 0.589·23-s + 0.138·25-s + 0.235·29-s − 0.134i·31-s + 0.403i·35-s + 1.14i·37-s + 0.867i·41-s − 1.32·43-s + 1.59·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.965 + 0.258i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.965 + 0.258i)$
$L(1)$  $\approx$  $1.099030517$
$L(\frac12)$  $\approx$  $1.099030517$
$L(\frac{3}{2})$   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 \( 1 + 2.38T + 5T^{2} \)
11 \( 1 - 1.65iT - 11T^{2} \)
13 \( 1 - 0.253iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 7.03T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 + 0.746iT - 31T^{2} \)
37 \( 1 - 6.94iT - 37T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 + 8.67T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 - 4.53iT - 61T^{2} \)
67 \( 1 + 0.253T + 67T^{2} \)
71 \( 1 - 3.17T + 71T^{2} \)
73 \( 1 + 3.05T + 73T^{2} \)
79 \( 1 + 14.5iT - 79T^{2} \)
83 \( 1 + 1.77iT - 83T^{2} \)
89 \( 1 + 3.13iT - 89T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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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

−8.083679067137710054894274687669, −7.30178780824970163248349817780, −6.79888101644030791953918322030, −6.04022650653550618500942447924, −4.87680711370119957624999130869, −4.44468225215080083094621476762, −3.69779969072359597256443424543, −2.84854939083634840573830341900, −1.78649613979293731993243945337, −0.50314665263146546899600447676, 0.57624787825082764553617777649, 1.95590941314841440997262091269, 2.87099715177283997738541561912, 3.85184072308218260393798588702, 4.23078774854954371359444509129, 5.27644143235792880972701729441, 5.95882970220985249427372870292, 6.77891507493532385183967446923, 7.40124296740184832766202033517, 8.285750141447135330333890254364

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