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  − 0.517i·5-s + i·7-s − 5.63·11-s + 1.91·13-s + 1.29i·17-s + 4.23i·19-s − 4.38·23-s + 4.73·25-s − 3.70i·29-s + 5.11i·31-s + 0.517·35-s + 2.18·37-s + 1.88i·41-s − 5.37i·43-s + 8.32·47-s + ⋯
L(s)  = 1  − 0.231i·5-s + 0.377i·7-s − 1.69·11-s + 0.531·13-s + 0.314i·17-s + 0.971i·19-s − 0.913·23-s + 0.946·25-s − 0.687i·29-s + 0.918i·31-s + 0.0874·35-s + 0.358·37-s + 0.294i·41-s − 0.820i·43-s + 1.21·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$
$L(1)$  $\approx$  $0.4445294439$
$L(\frac12)$  $\approx$  $0.4445294439$
$L(\frac{3}{2})$   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 \( 1 + 0.517iT - 5T^{2} \)
11 \( 1 + 5.63T + 11T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
17 \( 1 - 1.29iT - 17T^{2} \)
19 \( 1 - 4.23iT - 19T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 + 3.70iT - 29T^{2} \)
31 \( 1 - 5.11iT - 31T^{2} \)
37 \( 1 - 2.18T + 37T^{2} \)
41 \( 1 - 1.88iT - 41T^{2} \)
43 \( 1 + 5.37iT - 43T^{2} \)
47 \( 1 - 8.32T + 47T^{2} \)
53 \( 1 + 5.53iT - 53T^{2} \)
59 \( 1 - 2.54T + 59T^{2} \)
61 \( 1 + 0.0617T + 61T^{2} \)
67 \( 1 + 7.31iT - 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 16.9iT - 79T^{2} \)
83 \( 1 + 8.59T + 83T^{2} \)
89 \( 1 + 1.33iT - 89T^{2} \)
97 \( 1 - 5.23T + 97T^{2} \)
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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

−7.87499858599900015530119160029, −7.26140628722161556020186119547, −6.18026472438797173612321594602, −5.71356295598202818101767776170, −5.00127392019765020110644337439, −4.19051523673267322356799381432, −3.24880129878797165156143738694, −2.46329843443583406376166281684, −1.52626658171478029599031979934, −0.11950987000577543486535294567, 1.07775470928758121181908565081, 2.47005909837063214453454473324, 2.89852429814616581135566335266, 4.00905497570637354443008705255, 4.73157282929139268921383665794, 5.50626384814095076794373743155, 6.14804417737878274444873919627, 7.14204368453501111265372994403, 7.49969017328417078965036981837, 8.317553605599352512424496734467

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