Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.951 + 0.309i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0549i·5-s + 7-s − 2.63i·11-s − 3.67i·13-s − 3.16·17-s − 3.07i·19-s + 2.86·23-s + 4.99·25-s − 10.1i·29-s − 9.32·31-s − 0.0549i·35-s + 0.774i·37-s − 6.36·41-s + 9.98i·43-s − 12.3·47-s + ⋯
L(s)  = 1  − 0.0245i·5-s + 0.377·7-s − 0.794i·11-s − 1.02i·13-s − 0.766·17-s − 0.706i·19-s + 0.598·23-s + 0.999·25-s − 1.89i·29-s − 1.67·31-s − 0.00928i·35-s + 0.127i·37-s − 0.993·41-s + 1.52i·43-s − 1.79·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.951 + 0.309i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.951 + 0.309i)$
$L(1)$  $\approx$  $0.8235977679$
$L(\frac12)$  $\approx$  $0.8235977679$
$L(\frac{3}{2})$   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 \( 1 + 0.0549iT - 5T^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
13 \( 1 + 3.67iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 3.07iT - 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + 10.1iT - 29T^{2} \)
31 \( 1 + 9.32T + 31T^{2} \)
37 \( 1 - 0.774iT - 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 - 9.98iT - 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 3.39iT - 53T^{2} \)
59 \( 1 - 6.93iT - 59T^{2} \)
61 \( 1 + 8.35iT - 61T^{2} \)
67 \( 1 - 8.93iT - 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 - 8.38T + 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.964914784555984297831047783435, −6.98293900763097203116516042113, −6.39857734709132142641115971405, −5.51381807289873981687954792219, −4.96684290402380437708012803907, −4.10721521122838354931210469149, −3.16529628961934030295718698536, −2.50300254509273162789504042894, −1.28723452011757111973048910573, −0.20408788389810471514423356198, 1.52577281624139652726553959719, 2.04155863156007886665344119787, 3.27519964369773269051505457577, 3.99637315031583996782845618752, 4.96870912474217138260323122866, 5.23872542791739960039368156432, 6.55692765698626865290713156317, 6.90415579610635902759069615714, 7.54876110763442011950150837785, 8.574788397046003630326604577013

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