Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.389 - 0.920i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.61·5-s i·7-s + 2.00i·11-s + 1.59i·13-s + 7.79i·17-s + 5.05·19-s − 7.14·23-s + 8.03·25-s + 4.94·29-s + 2.53i·31-s − 3.61i·35-s + 4.76i·37-s − 0.836i·41-s − 0.151·43-s − 0.795·47-s + ⋯
L(s)  = 1  + 1.61·5-s − 0.377i·7-s + 0.605i·11-s + 0.441i·13-s + 1.88i·17-s + 1.15·19-s − 1.49·23-s + 1.60·25-s + 0.917·29-s + 0.454i·31-s − 0.610i·35-s + 0.783i·37-s − 0.130i·41-s − 0.0230·43-s − 0.116·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.389 - 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.389 - 0.920i)$
$L(1)$  $\approx$  $2.630711116$
$L(\frac12)$  $\approx$  $2.630711116$
$L(\frac{3}{2})$   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 \( 1 - 3.61T + 5T^{2} \)
11 \( 1 - 2.00iT - 11T^{2} \)
13 \( 1 - 1.59iT - 13T^{2} \)
17 \( 1 - 7.79iT - 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 + 7.14T + 23T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 - 4.76iT - 37T^{2} \)
41 \( 1 + 0.836iT - 41T^{2} \)
43 \( 1 + 0.151T + 43T^{2} \)
47 \( 1 + 0.795T + 47T^{2} \)
53 \( 1 - 1.05T + 53T^{2} \)
59 \( 1 + 8.34iT - 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 + 0.655T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 3.81iT - 79T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 + 0.0444iT - 89T^{2} \)
97 \( 1 - 9.86T + 97T^{2} \)
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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

−8.273538775212419132438981516322, −7.44973862830171642492640722700, −6.59046201772582759370033347175, −6.12002040035411694765267477743, −5.48650398533776501641467653698, −4.62844437224244174726224388692, −3.85131638942253586138738652031, −2.80129918553484670825594073639, −1.83457163858762806387426647899, −1.35900259238714226761576373542, 0.63407797298917511153003753469, 1.74190331478838623891951581651, 2.65231534152723635057183922523, 3.14775117189998346708542551823, 4.48388203199600231545949287199, 5.32191882986628644499690300588, 5.74669530318515127967337105438, 6.32329410034360214530431383649, 7.22842662121648961685697534538, 7.894335755418619863768119985945

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