Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.222 - 0.974i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.114i·5-s + 7-s − 0.412i·11-s + 1.73i·13-s − 2.50·17-s + 6.85i·19-s + 4.42·23-s + 4.98·25-s + 1.85i·29-s − 5.60·31-s + 0.114i·35-s − 4.39i·37-s + 2.39·41-s + 4.35i·43-s − 7.23·47-s + ⋯
L(s)  = 1  + 0.0512i·5-s + 0.377·7-s − 0.124i·11-s + 0.481i·13-s − 0.608·17-s + 1.57i·19-s + 0.922·23-s + 0.997·25-s + 0.344i·29-s − 1.00·31-s + 0.0193i·35-s − 0.721i·37-s + 0.374·41-s + 0.664i·43-s − 1.05·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.222 - 0.974i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.222 - 0.974i)$
$L(1)$  $\approx$  $1.453095890$
$L(\frac12)$  $\approx$  $1.453095890$
$L(\frac{3}{2})$   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 \( 1 - T \)
good5 \( 1 - 0.114iT - 5T^{2} \)
11 \( 1 + 0.412iT - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + 2.50T + 17T^{2} \)
19 \( 1 - 6.85iT - 19T^{2} \)
23 \( 1 - 4.42T + 23T^{2} \)
29 \( 1 - 1.85iT - 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 + 4.39iT - 37T^{2} \)
41 \( 1 - 2.39T + 41T^{2} \)
43 \( 1 - 4.35iT - 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 4.25iT - 59T^{2} \)
61 \( 1 - 7.35iT - 61T^{2} \)
67 \( 1 - 6.25iT - 67T^{2} \)
71 \( 1 + 0.608T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 8.19T + 79T^{2} \)
83 \( 1 - 4.88iT - 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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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

−8.395765233453716113402558311723, −7.47589318529275186482122834528, −6.96013433773805684853095371766, −6.13415648941194440536209235769, −5.41981762526910759181699484938, −4.66056860240950313990805979864, −3.89229760011228613951641480683, −3.07433880481644662185977318269, −2.04015607904672850785138738277, −1.19260855309888481872266146145, 0.38181355290503168494738758838, 1.52838110028575547339440084388, 2.61391143144682879276139468239, 3.25464203810867344608758102379, 4.44656013757268849164322120110, 4.87834333218416258008120111381, 5.65053442337013276227948863086, 6.61122923213135897971572076190, 7.11816439299611931709728576577, 7.83215094753807869843706644661

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