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  − 2i·5-s + 7-s + 5i·13-s − 17-s − 4i·19-s − 5·23-s + 25-s + 9i·29-s + 7·31-s − 2i·35-s + 2i·37-s − 2·41-s − 5i·43-s + 49-s + 9i·53-s + ⋯
L(s)  = 1  − 0.894i·5-s + 0.377·7-s + 1.38i·13-s − 0.242·17-s − 0.917i·19-s − 1.04·23-s + 0.200·25-s + 1.67i·29-s + 1.25·31-s − 0.338i·35-s + 0.328i·37-s − 0.312·41-s − 0.762i·43-s + 0.142·49-s + 1.23i·53-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  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.692702341$
$L(\frac12)$  $\approx$  $1.692702341$
$L(\frac{3}{2})$   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 \( 1 + 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 13T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.377838057258067812526962422941, −7.39351405520343852052235233563, −6.80673350326267440212694135962, −6.04814665091131104982570774359, −5.08990941333837346292580766516, −4.60778178362624913894369933707, −3.97668322015864853930066019495, −2.80647286765662166920991415580, −1.84123459574064292768280275779, −1.00138193297008511099446988485, 0.48488818372448211686100751049, 1.85179826352657479286814848016, 2.71974295480374889164080691186, 3.44258532200000085710139868701, 4.28808132185903897877814319405, 5.15615997851308991896631190871, 6.04250512282275106407261646131, 6.39649064168700851618727296672, 7.41189322615065550503117546900, 8.052005362050497021357290274740

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