Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.629 - 0.776i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.58i·5-s − 7-s + 0.790i·11-s + 0.494i·13-s − 4.46·17-s − 7.55i·19-s + 0.839·23-s + 2.48·25-s + 2.76i·29-s + 0.568·31-s − 1.58i·35-s + 0.343i·37-s + 5.93·41-s − 3.16i·43-s + 4.80·47-s + ⋯
L(s)  = 1  + 0.708i·5-s − 0.377·7-s + 0.238i·11-s + 0.137i·13-s − 1.08·17-s − 1.73i·19-s + 0.175·23-s + 0.497·25-s + 0.513i·29-s + 0.102·31-s − 0.267i·35-s + 0.0564i·37-s + 0.926·41-s − 0.482i·43-s + 0.701·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.629 - 0.776i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.629 - 0.776i)$
$L(1)$  $\approx$  $1.600663702$
$L(\frac12)$  $\approx$  $1.600663702$
$L(\frac{3}{2})$   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 \( 1 - 1.58iT - 5T^{2} \)
11 \( 1 - 0.790iT - 11T^{2} \)
13 \( 1 - 0.494iT - 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 + 7.55iT - 19T^{2} \)
23 \( 1 - 0.839T + 23T^{2} \)
29 \( 1 - 2.76iT - 29T^{2} \)
31 \( 1 - 0.568T + 31T^{2} \)
37 \( 1 - 0.343iT - 37T^{2} \)
41 \( 1 - 5.93T + 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 + 4.36iT - 53T^{2} \)
59 \( 1 - 4.50iT - 59T^{2} \)
61 \( 1 - 5.40iT - 61T^{2} \)
67 \( 1 - 7.57iT - 67T^{2} \)
71 \( 1 - 9.52T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 + 7.30iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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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

−8.184900642484219340376704538996, −7.11914073658714352334878781847, −6.93237124856682055351384743551, −6.24504985044269282159730510262, −5.25117606280009473283019733804, −4.55662862589504539096554004496, −3.72636450679801537694263135281, −2.75506747842988269998559878056, −2.27636552126330435620473221295, −0.789636238363610403320788014861, 0.55545688929593120064673279021, 1.65059445942718988625445836743, 2.63152368694358757780294615407, 3.63333684981519697119432940709, 4.30463676412523097645012764755, 5.09750571463749261037443393110, 5.89093014975270527117936566679, 6.44317316881263847694428616192, 7.30403523123080901984632054070, 8.155952061646935005446364870503

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