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  − 1.93i·5-s i·7-s − 1.13·11-s − 3.19·13-s + 5.93i·17-s − 1.33i·19-s + 2.96·23-s + 1.26·25-s + 1.65i·29-s + 10.3i·31-s − 1.93·35-s + 0.538·37-s − 9.51i·41-s − 6.65i·43-s + 11.1·47-s + ⋯
L(s)  = 1  − 0.863i·5-s − 0.377i·7-s − 0.342·11-s − 0.885·13-s + 1.43i·17-s − 0.307i·19-s + 0.618·23-s + 0.253·25-s + 0.307i·29-s + 1.86i·31-s − 0.326·35-s + 0.0884·37-s − 1.48i·41-s − 1.01i·43-s + 1.62·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $1.712585449$
$L(\frac12)$  $\approx$  $1.712585449$
$L(\frac{3}{2})$   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 \( 1 + 1.93iT - 5T^{2} \)
11 \( 1 + 1.13T + 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 - 5.93iT - 17T^{2} \)
19 \( 1 + 1.33iT - 19T^{2} \)
23 \( 1 - 2.96T + 23T^{2} \)
29 \( 1 - 1.65iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 0.538T + 37T^{2} \)
41 \( 1 + 9.51iT - 41T^{2} \)
43 \( 1 + 6.65iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 1.68iT - 53T^{2} \)
59 \( 1 + 1.82T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 9.53iT - 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 0.319T + 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 - 2.33T + 97T^{2} \)
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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

−8.129094378370252215152342885373, −7.11813922648766084999234620142, −6.84011408910756652181286530331, −5.55514478846110043880877256234, −5.22411231245184600354024379273, −4.36476730115895982198340164317, −3.66708066731535395232615903725, −2.61993924522183007900263247780, −1.63084880231314225902820459323, −0.62142533459574712590356305878, 0.75798516337044596031206769850, 2.40411984852942158807685518903, 2.63263810518953924568862784577, 3.61996083807183632121696792194, 4.67738873595282304161727983463, 5.22601942421252707113918040009, 6.14933335739891774292399442046, 6.73947607080889619339750754022, 7.59038331097616361300686777456, 7.83961117716014731190941432572

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