Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·5-s + i·7-s + 0.635·11-s + 0.550·13-s − 3.14i·17-s − 4i·19-s + 3.14·23-s − 3.00·25-s + 5.19i·29-s + 10.7i·31-s − 2.82·35-s − 8.89·37-s + 3.46i·41-s + 0.550i·43-s + 1.27·47-s + ⋯
L(s)  = 1  + 1.26i·5-s + 0.377i·7-s + 0.191·11-s + 0.152·13-s − 0.763i·17-s − 0.917i·19-s + 0.656·23-s − 0.600·25-s + 0.964i·29-s + 1.93i·31-s − 0.478·35-s − 1.46·37-s + 0.541i·41-s + 0.0839i·43-s + 0.185·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.499737674$
$L(\frac12)$  $\approx$  $1.499737674$
$L(\frac{3}{2})$   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 \( 1 - 2.82iT - 5T^{2} \)
11 \( 1 - 0.635T + 11T^{2} \)
13 \( 1 - 0.550T + 13T^{2} \)
17 \( 1 + 3.14iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 3.14T + 23T^{2} \)
29 \( 1 - 5.19iT - 29T^{2} \)
31 \( 1 - 10.7iT - 31T^{2} \)
37 \( 1 + 8.89T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 0.550iT - 43T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 - 2.36iT - 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 3.10T + 61T^{2} \)
67 \( 1 - 1.44iT - 67T^{2} \)
71 \( 1 - 5.97T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 - 3.79T + 97T^{2} \)
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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

−8.481415624479944600497705563177, −7.33794454161017323428264815797, −6.89292910815822881335913721648, −6.52055482159140289676092288728, −5.35872944428126785828484365924, −4.93428243250227462729377489070, −3.70873379042775248869398326096, −3.03567225201574593517954454867, −2.45773615555051832536099576482, −1.22189054088878066310580453652, 0.40470572102400790697840739960, 1.37985690773701066659229118108, 2.23878206673743941876592801026, 3.63182429408756541869769349668, 4.08995529353660955133748715918, 4.91639334884192501965443769333, 5.64665271091894623485697654581, 6.25735012595234675247692646355, 7.21030559047475203227007091679, 7.933063578764701252261150879852

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