Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.587 - 0.809i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60i·5-s + 7-s + 0.0549i·11-s − 3.75i·13-s − 3.16·17-s + 0.726i·19-s − 7.77·23-s + 2.41·25-s − 2.75i·29-s + 2.14·31-s + 1.60i·35-s − 0.600i·37-s + 4.53·41-s + 6.85i·43-s − 0.744·47-s + ⋯
L(s)  = 1  + 0.718i·5-s + 0.377·7-s + 0.0165i·11-s − 1.04i·13-s − 0.766·17-s + 0.166i·19-s − 1.62·23-s + 0.483·25-s − 0.512i·29-s + 0.385·31-s + 0.271i·35-s − 0.0987i·37-s + 0.708·41-s + 1.04i·43-s − 0.108·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.587 - 0.809i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.587 - 0.809i)$
$L(1)$  $\approx$  $1.066872941$
$L(\frac12)$  $\approx$  $1.066872941$
$L(\frac{3}{2})$   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 \( 1 - 1.60iT - 5T^{2} \)
11 \( 1 - 0.0549iT - 11T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 - 0.726iT - 19T^{2} \)
23 \( 1 + 7.77T + 23T^{2} \)
29 \( 1 + 2.75iT - 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 + 0.600iT - 37T^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 - 6.85iT - 43T^{2} \)
47 \( 1 + 0.744T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 2.58iT - 59T^{2} \)
61 \( 1 - 9.80iT - 61T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 + 9.19T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 + 7.90T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.232317179632578937364679836209, −7.62604984405640838111369823162, −6.98160224135288284676494838869, −6.07395993559625087482062797194, −5.69419671781483677666026789029, −4.57328462841213539274018688763, −4.00585925864239316933270383596, −2.92189445262588551469934960940, −2.38240937646334405216427050374, −1.15479111511585884856525419584, 0.27540053526496939001118057155, 1.59857031474942986376177477844, 2.22914019484091124148329022816, 3.47045860698091191302265463758, 4.36400844002157529139168787413, 4.76894645765294032805142735000, 5.65485544556012692624732054599, 6.45418216119642479149176371695, 7.06925220576501175910738691067, 7.967346471280940399344874008785

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