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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.438994664$
$L(\frac12)$  $\approx$  $2.438994664$
$L(\frac{3}{2})$   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

−8.264171191602131491254014590472, −7.33935604202371017185813379468, −6.65246019649911300583156583831, −6.22039765387463484286863386864, −5.29569728721254105483534794560, −4.54180134412750086189025089044, −3.59058753528493698231950568948, −3.10338363555786948246002180601, −1.81552149453283991106919097199, −1.05682590600848104032037128752, 0.77063613472039240075296921307, 1.50767556363360682612811344730, 2.73499446850814366572809617864, 3.64917764487856473079897451004, 4.31258612190528184280019376290, 4.97308394999107413175113776547, 6.02147218494099228039500986031, 6.61747530279755952482660097573, 7.10349415907982869246731453004, 8.102654636167769515871886440863

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