Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.880 - 0.474i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.66i·5-s − 7-s − 4.45i·11-s + 1.51i·13-s + 3.45·17-s − 2.29i·19-s − 8.76·23-s − 8.44·25-s − 1.62i·29-s + 9.26·31-s − 3.66i·35-s − 5.69i·37-s + 6.86·41-s − 0.880i·43-s + 10.5·47-s + ⋯
L(s)  = 1  + 1.63i·5-s − 0.377·7-s − 1.34i·11-s + 0.420i·13-s + 0.837·17-s − 0.527i·19-s − 1.82·23-s − 1.68·25-s − 0.300i·29-s + 1.66·31-s − 0.619i·35-s − 0.936i·37-s + 1.07·41-s − 0.134i·43-s + 1.54·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.880 - 0.474i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.880 - 0.474i)$
$L(1)$  $\approx$  $1.762482639$
$L(\frac12)$  $\approx$  $1.762482639$
$L(\frac{3}{2})$   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 \( 1 - 3.66iT - 5T^{2} \)
11 \( 1 + 4.45iT - 11T^{2} \)
13 \( 1 - 1.51iT - 13T^{2} \)
17 \( 1 - 3.45T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + 8.76T + 23T^{2} \)
29 \( 1 + 1.62iT - 29T^{2} \)
31 \( 1 - 9.26T + 31T^{2} \)
37 \( 1 + 5.69iT - 37T^{2} \)
41 \( 1 - 6.86T + 41T^{2} \)
43 \( 1 + 0.880iT - 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 2.99iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 + 4.57T + 79T^{2} \)
83 \( 1 + 2.60iT - 83T^{2} \)
89 \( 1 - 2.14T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.88684213614477831501429618218, −7.51819722844721023174292620122, −6.50930827862753573071634846465, −6.18244356128816841861956449536, −5.58107710990254049307458651418, −4.23069772620005804090583702356, −3.59690730271157781807895586632, −2.85580589657525352861704120829, −2.24057394397914485207285970609, −0.66839369031574198395236030177, 0.73506148102357086315235326241, 1.62631291248241466036580854988, 2.58010781633058228569842974305, 3.87299444285027678002881751001, 4.35488911448510869006824449559, 5.15361190429624794337251661488, 5.73361383876385395905740016808, 6.52583627035529905933359773579, 7.52996237261716999498507110518, 8.086371391866137167547933991519

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