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  + 2i·5-s + 7-s + 5i·13-s + 17-s − 4i·19-s + 5·23-s + 25-s − 9i·29-s + 7·31-s + 2i·35-s + 2i·37-s + 2·41-s − 5i·43-s + 49-s − 9i·53-s + ⋯
L(s)  = 1  + 0.894i·5-s + 0.377·7-s + 1.38i·13-s + 0.242·17-s − 0.917i·19-s + 1.04·23-s + 0.200·25-s − 1.67i·29-s + 1.25·31-s + 0.338i·35-s + 0.328i·37-s + 0.312·41-s − 0.762i·43-s + 0.142·49-s − 1.23i·53-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  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.243389270$
$L(\frac12)$  $\approx$  $2.243389270$
$L(\frac{3}{2})$   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 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 5T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 13T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.160121445874212267566844211224, −7.29296081509303595524054842237, −6.75885488534285347256875211226, −6.27963530035108298059024491788, −5.21006649429812336758834395192, −4.52744584124502477203492833870, −3.76819757261829282929161123519, −2.75163516912380957984230910762, −2.16884720685062768093969796126, −0.890840586339115948092554026183, 0.78187672482828431219914192437, 1.46644659443831966827454615355, 2.77789726791239328209640723026, 3.46982435331385974920950612120, 4.53267727581114999926143974025, 5.12834744773377447375616099912, 5.64927597714867348917153816715, 6.55062231685487687322961879923, 7.43015231496483898726799062342, 8.129666030689296445274998829153

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