Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.0430 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.11i·5-s − 7-s + 2.37i·11-s + 1.09i·13-s + 3.69·17-s − 1.08i·19-s − 4.87·23-s − 4.69·25-s − 1.59i·29-s − 7.45·31-s + 3.11i·35-s − 4.61i·37-s − 0.0380·41-s + 11.0i·43-s − 0.337·47-s + ⋯
L(s)  = 1  − 1.39i·5-s − 0.377·7-s + 0.715i·11-s + 0.302i·13-s + 0.895·17-s − 0.248i·19-s − 1.01·23-s − 0.939·25-s − 0.296i·29-s − 1.33·31-s + 0.526i·35-s − 0.758i·37-s − 0.00594·41-s + 1.68i·43-s − 0.0491·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0430 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0430 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.0430 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.0430 - 0.999i)$
$L(1)$  $\approx$  $0.7642840685$
$L(\frac12)$  $\approx$  $0.7642840685$
$L(\frac{3}{2})$   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 \( 1 + 3.11iT - 5T^{2} \)
11 \( 1 - 2.37iT - 11T^{2} \)
13 \( 1 - 1.09iT - 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 + 1.08iT - 19T^{2} \)
23 \( 1 + 4.87T + 23T^{2} \)
29 \( 1 + 1.59iT - 29T^{2} \)
31 \( 1 + 7.45T + 31T^{2} \)
37 \( 1 + 4.61iT - 37T^{2} \)
41 \( 1 + 0.0380T + 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + 0.337T + 47T^{2} \)
53 \( 1 - 8.14iT - 53T^{2} \)
59 \( 1 - 15.0iT - 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 + 7.70iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 5.30T + 97T^{2} \)
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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

−8.238611943992951266368427256317, −7.61633433911759707411404454065, −6.94045573742407206920537436356, −5.85114632776293650421224980470, −5.50834074203584264190434170269, −4.46123214668095912400075296414, −4.14426718202127354702885706702, −3.00172853691413380287350685750, −1.90201519374852987342094639195, −1.07452702839921028914304314145, 0.20471044139538925537361348673, 1.73192316738872415306734782096, 2.72947992555364940246738069246, 3.44650500469602123125135034917, 3.87010687911252628418877349182, 5.29424788686246327947955179861, 5.78193162649099072323444749886, 6.60834172732248166906354314319, 7.05578723092797499128384630747, 7.897804046003496604973218528586

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