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  + 2.82i·5-s i·7-s + 6.29·11-s + 5.44·13-s + 0.317i·17-s + 4i·19-s + 0.317·23-s − 3.00·25-s − 5.19i·29-s + 8.79i·31-s + 2.82·35-s + 0.898·37-s − 3.46i·41-s − 5.44i·43-s + 12.5·47-s + ⋯
L(s)  = 1  + 1.26i·5-s − 0.377i·7-s + 1.89·11-s + 1.51·13-s + 0.0770i·17-s + 0.917i·19-s + 0.0662·23-s − 0.600·25-s − 0.964i·29-s + 1.58i·31-s + 0.478·35-s + 0.147·37-s − 0.541i·41-s − 0.831i·43-s + 1.83·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.676032112$
$L(\frac12)$  $\approx$  $2.676032112$
$L(\frac{3}{2})$   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 \( 1 - 2.82iT - 5T^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 - 5.44T + 13T^{2} \)
17 \( 1 - 0.317iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 0.317T + 23T^{2} \)
29 \( 1 + 5.19iT - 29T^{2} \)
31 \( 1 - 8.79iT - 31T^{2} \)
37 \( 1 - 0.898T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 5.44iT - 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + 8.02iT - 53T^{2} \)
59 \( 1 - 5.83T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 3.44iT - 67T^{2} \)
71 \( 1 + 2.51T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.262732675009749924311405810538, −7.13139108442170507845688079179, −6.83258431261477919026930789592, −6.17729439047123536690708651385, −5.56254526639074086944540018322, −4.05915837708258683585081516155, −3.86360052386991843675545953188, −3.08639485708398051626228762603, −1.88609004962703733541430868713, −1.02561594894084176741256905138, 0.938969031279005878031023495077, 1.36215721285144881376278606286, 2.62461150650587471012642969032, 3.89099480478945901539650550197, 4.15186423508217935588056707225, 5.12846551602665574758799404523, 5.87732092465968555299478993120, 6.46357803973003797395048426618, 7.21611848341436439876199832374, 8.322094304540394523359883910132

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