Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.600 + 0.799i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.940i·5-s − 7-s + 5.98i·11-s − 6.59i·13-s − 2.64·17-s + 5.83i·19-s − 2.88·23-s + 4.11·25-s + 3.09i·29-s − 3.52·31-s + 0.940i·35-s − 0.213i·37-s − 1.63·41-s − 7.16i·43-s + 9.32·47-s + ⋯
L(s)  = 1  − 0.420i·5-s − 0.377·7-s + 1.80i·11-s − 1.82i·13-s − 0.640·17-s + 1.33i·19-s − 0.601·23-s + 0.823·25-s + 0.575i·29-s − 0.632·31-s + 0.158i·35-s − 0.0351i·37-s − 0.255·41-s − 1.09i·43-s + 1.36·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.600 + 0.799i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.600 + 0.799i)$
$L(1)$  $\approx$  $0.7337947644$
$L(\frac12)$  $\approx$  $0.7337947644$
$L(\frac{3}{2})$   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 \( 1 + 0.940iT - 5T^{2} \)
11 \( 1 - 5.98iT - 11T^{2} \)
13 \( 1 + 6.59iT - 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 - 5.83iT - 19T^{2} \)
23 \( 1 + 2.88T + 23T^{2} \)
29 \( 1 - 3.09iT - 29T^{2} \)
31 \( 1 + 3.52T + 31T^{2} \)
37 \( 1 + 0.213iT - 37T^{2} \)
41 \( 1 + 1.63T + 41T^{2} \)
43 \( 1 + 7.16iT - 43T^{2} \)
47 \( 1 - 9.32T + 47T^{2} \)
53 \( 1 + 7.51iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 1.48iT - 61T^{2} \)
67 \( 1 + 13.0iT - 67T^{2} \)
71 \( 1 - 1.54T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 - 8.74iT - 83T^{2} \)
89 \( 1 - 7.50T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.80516159529568812735307595821, −7.19596354184515089213198740674, −6.46009790278360116593259059024, −5.51365065384666571204443218904, −5.07049905184906369283746486577, −4.14335349680272156224743303049, −3.41965437253325142243833808219, −2.38602262604711252945416539762, −1.54095942128619160383034740016, −0.19878271310434394185993081777, 1.07729777590360392172838580286, 2.38689943519390139789982789313, 2.98826942310227399503666454901, 3.98415777453166174503844772429, 4.53625818901723936804232099519, 5.66447143297450662960445025050, 6.26821977740282897972971622467, 6.84150569438240520102210855824, 7.45181477067709864207595398164, 8.567842896878346323400646741928

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