Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0820 + 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.99·5-s + i·7-s − 5.36i·11-s − 4.55i·13-s + 0.958i·17-s − 0.532·19-s − 2.78·23-s + 3.98·25-s + 5.15·29-s − 4.66i·31-s + 2.99i·35-s + 6.10i·37-s − 12.3i·41-s − 4.45·43-s − 1.40·47-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377i·7-s − 1.61i·11-s − 1.26i·13-s + 0.232i·17-s − 0.122·19-s − 0.581·23-s + 0.796·25-s + 0.957·29-s − 0.837i·31-s + 0.506i·35-s + 1.00i·37-s − 1.92i·41-s − 0.679·43-s − 0.204·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0820 + 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0820 + 0.996i)$
$L(1)$  $\approx$  $2.161595780$
$L(\frac12)$  $\approx$  $2.161595780$
$L(\frac{3}{2})$   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 \( 1 - iT \)
good5 \( 1 - 2.99T + 5T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 + 4.55iT - 13T^{2} \)
17 \( 1 - 0.958iT - 17T^{2} \)
19 \( 1 + 0.532T + 19T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 - 5.15T + 29T^{2} \)
31 \( 1 + 4.66iT - 31T^{2} \)
37 \( 1 - 6.10iT - 37T^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + 4.45T + 43T^{2} \)
47 \( 1 + 1.40T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 + 0.0356iT - 59T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 + 8.59T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 - 1.90iT - 79T^{2} \)
83 \( 1 + 3.13iT - 83T^{2} \)
89 \( 1 - 5.91iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−8.221161859789931289300821676997, −7.06716560493345098624619879396, −6.22118536951370206179591198215, −5.64624272178870965444499070434, −5.47356442158999619914278585794, −4.22761468770874143019794524195, −3.17360710064668722229919400169, −2.63083902808365771249140418368, −1.60909861293463993074218214975, −0.50881602016932560952981729567, 1.40261656667593405839577957726, 1.97859064588039362045976005460, 2.77582935099301762956364191826, 4.06776974500403376533744816542, 4.65832704388081331636412144802, 5.33344020735079452794577165221, 6.34073244309711289888454695887, 6.71777776188891875773217950409, 7.39097208445382348847142162143, 8.277949488131681618373461156922

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