Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.534 + 0.845i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.16i·5-s + 7-s + 5.44i·11-s + 3.61i·13-s − 3.27·17-s − 3.20i·19-s + 0.673·23-s − 5.04·25-s + 2.85i·29-s − 3.71·31-s − 3.16i·35-s − 11.9i·37-s + 7.44·41-s − 12.5i·43-s − 4.06·47-s + ⋯
L(s)  = 1  − 1.41i·5-s + 0.377·7-s + 1.64i·11-s + 1.00i·13-s − 0.794·17-s − 0.735i·19-s + 0.140·23-s − 1.00·25-s + 0.529i·29-s − 0.667·31-s − 0.535i·35-s − 1.97i·37-s + 1.16·41-s − 1.91i·43-s − 0.592·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.534 + 0.845i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.534 + 0.845i)$
$L(1)$  $\approx$  $1.808547831$
$L(\frac12)$  $\approx$  $1.808547831$
$L(\frac{3}{2})$   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 \( 1 + 3.16iT - 5T^{2} \)
11 \( 1 - 5.44iT - 11T^{2} \)
13 \( 1 - 3.61iT - 13T^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 + 3.20iT - 19T^{2} \)
23 \( 1 - 0.673T + 23T^{2} \)
29 \( 1 - 2.85iT - 29T^{2} \)
31 \( 1 + 3.71T + 31T^{2} \)
37 \( 1 + 11.9iT - 37T^{2} \)
41 \( 1 - 7.44T + 41T^{2} \)
43 \( 1 + 12.5iT - 43T^{2} \)
47 \( 1 + 4.06T + 47T^{2} \)
53 \( 1 - 0.291iT - 53T^{2} \)
59 \( 1 + 0.0209iT - 59T^{2} \)
61 \( 1 + 5.34iT - 61T^{2} \)
67 \( 1 - 6.20iT - 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 1.35T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 13.6iT - 83T^{2} \)
89 \( 1 - 5.93T + 89T^{2} \)
97 \( 1 - 9.60T + 97T^{2} \)
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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

−7.998145923576016018688469265306, −7.16809607474331218507460775008, −6.79122838902687277370336040254, −5.56685865457164484222673170023, −4.97888947438283762128096712176, −4.41279009639881240258778434804, −3.86999900917113686739466322528, −2.19390002763051045746961623566, −1.85158276751386797058263131351, −0.57909269510595634065377613383, 0.857668939277944377439496754356, 2.18905236669551308591336683114, 3.12792172772313041233700719867, 3.40455554822442508712952945862, 4.53913180348086213302932622103, 5.49531739439025032217401313998, 6.26157250873564357787569723177, 6.51844034314124381650515270641, 7.70202896696784233337290265970, 7.971864268452896617811147667019

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