Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0841 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46i·5-s − 7-s − 3.31i·11-s − 3.10i·13-s − 1.40·17-s + 4.80i·19-s − 8.79·23-s − 6.99·25-s + 9.87i·29-s + 7.83·31-s + 3.46i·35-s − 5.42i·37-s − 11.5·41-s + 7.03i·43-s − 11.2·47-s + ⋯
L(s)  = 1  − 1.54i·5-s − 0.377·7-s − 1.00i·11-s − 0.862i·13-s − 0.340·17-s + 1.10i·19-s − 1.83·23-s − 1.39·25-s + 1.83i·29-s + 1.40·31-s + 0.585i·35-s − 0.891i·37-s − 1.80·41-s + 1.07i·43-s − 1.64·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0841 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0841 - 0.996i)$
$L(1)$  $\approx$  $0.1284181636$
$L(\frac12)$  $\approx$  $0.1284181636$
$L(\frac{3}{2})$   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 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + 3.31iT - 11T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 - 4.80iT - 19T^{2} \)
23 \( 1 + 8.79T + 23T^{2} \)
29 \( 1 - 9.87iT - 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 + 5.42iT - 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 7.03iT - 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + 6.51iT - 53T^{2} \)
59 \( 1 + 3.89iT - 59T^{2} \)
61 \( 1 - 9.42iT - 61T^{2} \)
67 \( 1 + 0.909iT - 67T^{2} \)
71 \( 1 - 6.42T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 0.370iT - 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−8.099248021732456278748026967184, −8.085653385735728000510070797645, −6.69172438619381469705982458802, −6.02073519200834462002136131662, −5.36990009634371323158104395928, −4.78042440658428065212047951347, −3.80330426584092474486455147774, −3.21615100795973700706195336373, −1.89377963050074349213719889046, −1.00077602996809769598294945907, 0.03490738073083201860118625796, 1.95364650396496352615502571638, 2.45146797888473917363368237887, 3.36770603786770231006670321462, 4.19270096931919908117392773903, 4.85004619147125529753434033495, 6.11465471235099183549130646197, 6.58370666543008753645378767009, 6.94976305775526383474054784198, 7.80509584684739649973320202987

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