Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.399 + 0.916i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53i·5-s − 7-s − 2.28i·11-s − 7.10i·13-s + 6.81·17-s − 1.60i·19-s − 1.16·23-s + 2.64·25-s − 4.07i·29-s + 7.90·31-s + 1.53i·35-s + 7.04i·37-s + 10.1·41-s + 0.344i·43-s − 3.10·47-s + ⋯
L(s)  = 1  − 0.685i·5-s − 0.377·7-s − 0.688i·11-s − 1.97i·13-s + 1.65·17-s − 0.368i·19-s − 0.243·23-s + 0.529·25-s − 0.755i·29-s + 1.41·31-s + 0.259i·35-s + 1.15i·37-s + 1.58·41-s + 0.0525i·43-s − 0.453·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.399 + 0.916i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.399 + 0.916i)$
$L(1)$  $\approx$  $1.905035520$
$L(\frac12)$  $\approx$  $1.905035520$
$L(\frac{3}{2})$   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 \( 1 + 1.53iT - 5T^{2} \)
11 \( 1 + 2.28iT - 11T^{2} \)
13 \( 1 + 7.10iT - 13T^{2} \)
17 \( 1 - 6.81T + 17T^{2} \)
19 \( 1 + 1.60iT - 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 4.07iT - 29T^{2} \)
31 \( 1 - 7.90T + 31T^{2} \)
37 \( 1 - 7.04iT - 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 0.344iT - 43T^{2} \)
47 \( 1 + 3.10T + 47T^{2} \)
53 \( 1 - 5.66iT - 53T^{2} \)
59 \( 1 + 1.38iT - 59T^{2} \)
61 \( 1 + 5.50iT - 61T^{2} \)
67 \( 1 - 8.35iT - 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 2.60T + 97T^{2} \)
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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

−8.111399450257011173539533867234, −7.30476964401538918739366647233, −6.17841676476322230704528905405, −5.75049273614323031322197413146, −5.07679448738036713728907689024, −4.23913367214257473386102638918, −3.08848959020706805973234391881, −2.88958921084509193326643488479, −1.17105536473991576840618359837, −0.57928690348246487872520001031, 1.21730649075790780814725802006, 2.19778226827309367266473551658, 3.04945051260353620571776760535, 3.91171453173403582186496127861, 4.55492420576931800412628938445, 5.52532141021502877005758902326, 6.31909918841194820831097461998, 6.90135245890128447960318255339, 7.42849975157135729735679075697, 8.212108408497359925675264322053

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