Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.258 + 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.12·5-s i·7-s + 3.86i·11-s + 5.08i·13-s − 2i·17-s − 7.99·19-s − 7.68·23-s − 3.72·25-s + 4.73·29-s + 4.08i·31-s + 1.12i·35-s − 8.28i·37-s − 8.41i·41-s + 10.0·43-s − 1.93·47-s + ⋯
L(s)  = 1  − 0.505·5-s − 0.377i·7-s + 1.16i·11-s + 1.41i·13-s − 0.485i·17-s − 1.83·19-s − 1.60·23-s − 0.744·25-s + 0.878·29-s + 0.733i·31-s + 0.190i·35-s − 1.36i·37-s − 1.31i·41-s + 1.52·43-s − 0.282·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.258 + 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.258 + 0.965i)$
$L(1)$  $\approx$  $0.8069843359$
$L(\frac12)$  $\approx$  $0.8069843359$
$L(\frac{3}{2})$   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 \( 1 + 1.12T + 5T^{2} \)
11 \( 1 - 3.86iT - 11T^{2} \)
13 \( 1 - 5.08iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 7.99T + 19T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 4.08iT - 31T^{2} \)
37 \( 1 + 8.28iT - 37T^{2} \)
41 \( 1 + 8.41iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 1.93T + 47T^{2} \)
53 \( 1 - 7.72T + 53T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 + 5.08T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 2.63T + 73T^{2} \)
79 \( 1 + 7.17iT - 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 - 4.17T + 97T^{2} \)
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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

−7.75316538876823892317601122608, −7.28118725990732990531663350183, −6.59776913423956655089710939633, −5.93952445722253791854862366773, −4.78252464010910632731486021199, −4.17347389236523710713219962311, −3.84562625737940863111967052961, −2.29620108616098144077442724052, −1.88837341919346447752905858732, −0.26003176139400920359723001397, 0.806271150537698766524519076454, 2.18248130541583922713924155289, 2.97174111818743424554710859370, 3.85700850885857279857025776868, 4.43674044426087605977334432652, 5.58826379943648067549220115005, 6.03029960208503269942324297943, 6.62653603042762850213953856432, 7.940247504331194053100879028727, 8.127578042994167383708657265591

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