Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.222 - 0.974i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.114i·5-s + 7-s + 0.412i·11-s + 1.73i·13-s + 2.50·17-s + 6.85i·19-s − 4.42·23-s + 4.98·25-s − 1.85i·29-s − 5.60·31-s − 0.114i·35-s − 4.39i·37-s − 2.39·41-s + 4.35i·43-s + 7.23·47-s + ⋯
L(s)  = 1  − 0.0512i·5-s + 0.377·7-s + 0.124i·11-s + 0.481i·13-s + 0.608·17-s + 1.57i·19-s − 0.922·23-s + 0.997·25-s − 0.344i·29-s − 1.00·31-s − 0.0193i·35-s − 0.721i·37-s − 0.374·41-s + 0.664i·43-s + 1.05·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.222 - 0.974i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.222 - 0.974i)$
$L(1)$  $\approx$  $1.462261034$
$L(\frac12)$  $\approx$  $1.462261034$
$L(\frac{3}{2})$   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 \( 1 + 0.114iT - 5T^{2} \)
11 \( 1 - 0.412iT - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 - 2.50T + 17T^{2} \)
19 \( 1 - 6.85iT - 19T^{2} \)
23 \( 1 + 4.42T + 23T^{2} \)
29 \( 1 + 1.85iT - 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 + 4.39iT - 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 4.35iT - 43T^{2} \)
47 \( 1 - 7.23T + 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + 4.25iT - 59T^{2} \)
61 \( 1 - 7.35iT - 61T^{2} \)
67 \( 1 - 6.25iT - 67T^{2} \)
71 \( 1 - 0.608T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 8.19T + 79T^{2} \)
83 \( 1 + 4.88iT - 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 3.42T + 97T^{2} \)
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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

−8.267406995634446613572145738774, −7.54215294995282156659198429753, −7.02798799705246632092323335909, −5.85645170362532475654374870236, −5.69765532067549853730618519147, −4.51189541686781446744369742865, −3.99446658858580949974860336730, −3.06107389508538299237720090497, −2.00806395052403941760453756993, −1.21538550115896424608599754056, 0.38096039076953683893972454969, 1.53219022979678868185813734659, 2.59014465134489468388595444924, 3.33404579739376068442276156815, 4.25134174198101010050132318619, 5.09606146023762239277397141598, 5.57762490179219351833578652699, 6.56881386680323976712724086733, 7.16317484044592989789425886083, 7.85932342346967763186355694393

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