Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.331 + 0.943i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.47·5-s + i·7-s − 5.11i·11-s + 0.268i·13-s + 1.96i·17-s − 0.909·19-s − 5.86·23-s + 7.04·25-s − 7.19·29-s − 4.57i·31-s + 3.47i·35-s − 6.98i·37-s − 8.34i·41-s + 9.25·43-s + 6.49·47-s + ⋯
L(s)  = 1  + 1.55·5-s + 0.377i·7-s − 1.54i·11-s + 0.0744i·13-s + 0.476i·17-s − 0.208·19-s − 1.22·23-s + 1.40·25-s − 1.33·29-s − 0.821i·31-s + 0.586i·35-s − 1.14i·37-s − 1.30i·41-s + 1.41·43-s + 0.947·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.331 + 0.943i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.331 + 0.943i)$
$L(1)$  $\approx$  $2.336923913$
$L(\frac12)$  $\approx$  $2.336923913$
$L(\frac{3}{2})$   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 \( 1 - 3.47T + 5T^{2} \)
11 \( 1 + 5.11iT - 11T^{2} \)
13 \( 1 - 0.268iT - 13T^{2} \)
17 \( 1 - 1.96iT - 17T^{2} \)
19 \( 1 + 0.909T + 19T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + 7.19T + 29T^{2} \)
31 \( 1 + 4.57iT - 31T^{2} \)
37 \( 1 + 6.98iT - 37T^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 - 6.49T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 3.50iT - 59T^{2} \)
61 \( 1 + 1.96iT - 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + 9.01T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 4.96iT - 83T^{2} \)
89 \( 1 + 18.2iT - 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−8.012215192472246057696012676755, −7.19485072726364839570167581078, −6.14362849296650178206881372942, −5.79170619811245771021539635940, −5.52779452876710739416908412711, −4.21023217846903590130260966782, −3.45150567927644784982454399148, −2.34081810493461459394888487292, −1.90178841952352067613315707185, −0.56020638836161763193599954516, 1.24474889543280377245687268840, 2.04759227458208773932243794723, 2.66439047583172196799141234196, 3.92270871603351107504592699145, 4.67209641071654296983720762384, 5.40669745777288526758644085900, 6.05818850538427299330315398198, 6.80552684618022429831428311193, 7.36629900974614057918586949678, 8.167916326802079465281468054700

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