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  + 2.64i·5-s + i·7-s + 4.12·11-s − 4.86·13-s − 2.03i·17-s − 4.16i·19-s + 9.48·23-s − 1.99·25-s + 1.54i·29-s − 0.248i·31-s − 2.64·35-s + 7.88·37-s + 2.00i·41-s + 0.547i·43-s + 2.59·47-s + ⋯
L(s)  = 1  + 1.18i·5-s + 0.377i·7-s + 1.24·11-s − 1.35·13-s − 0.494i·17-s − 0.956i·19-s + 1.97·23-s − 0.398·25-s + 0.287i·29-s − 0.0446i·31-s − 0.446·35-s + 1.29·37-s + 0.313i·41-s + 0.0835i·43-s + 0.378·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.137287986$
$L(\frac12)$  $\approx$  $2.137287986$
$L(\frac{3}{2})$   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 \( 1 - 2.64iT - 5T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 + 2.03iT - 17T^{2} \)
19 \( 1 + 4.16iT - 19T^{2} \)
23 \( 1 - 9.48T + 23T^{2} \)
29 \( 1 - 1.54iT - 29T^{2} \)
31 \( 1 + 0.248iT - 31T^{2} \)
37 \( 1 - 7.88T + 37T^{2} \)
41 \( 1 - 2.00iT - 41T^{2} \)
43 \( 1 - 0.547iT - 43T^{2} \)
47 \( 1 - 2.59T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 6.52iT - 67T^{2} \)
71 \( 1 - 5.95T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 7.47iT - 79T^{2} \)
83 \( 1 - 3.94T + 83T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + 5.64T + 97T^{2} \)
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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

−8.064876662030712570076796037695, −7.13588875716192694312852962838, −6.90284985453768955709598563781, −6.28218566396175511304974728186, −5.16429586647966585104969046831, −4.67662526704622821681074581076, −3.54621368689525952962549086031, −2.84250630676006989671616221552, −2.23808657607683580958223795366, −0.831494375140135948996653399023, 0.78210144882727260029422571181, 1.47535502903929700656508948929, 2.62953484174176012983876429367, 3.73295728182039510306918095358, 4.41227260212813690106511718385, 4.99409025305640252894224937309, 5.77184070112609239664579347644, 6.63080081211503814223904111028, 7.30822562347185393236494183965, 8.007886939447452603896785690864

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