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Properties

Label 1-6048-6048.3461-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.0322 - 0.999i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.573 + 0.819i)5-s + (0.819 − 0.573i)11-s + (−0.819 − 0.573i)13-s + (0.5 − 0.866i)17-s + (0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.573 − 0.819i)29-s + (−0.173 − 0.984i)31-s + (0.707 + 0.707i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.173 + 0.984i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 + 0.819i)5-s + (0.819 − 0.573i)11-s + (−0.819 − 0.573i)13-s + (0.5 − 0.866i)17-s + (0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.573 − 0.819i)29-s + (−0.173 − 0.984i)31-s + (0.707 + 0.707i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.173 + 0.984i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.0322 - 0.999i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.0322 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8581029271 - 0.8308977540i\)
\(L(\frac12)\) \(\approx\) \(0.8581029271 - 0.8308977540i\)
\(L(1)\) \(\approx\) \(0.9561581779 - 0.06015405879i\)
\(L(1)\) \(\approx\) \(0.9561581779 - 0.06015405879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.573 + 0.819i)T \)
11 \( 1 + (0.819 - 0.573i)T \)
13 \( 1 + (-0.819 - 0.573i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.573 - 0.819i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.0871 - 0.996i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (-0.422 - 0.906i)T \)
61 \( 1 + (-0.573 - 0.819i)T \)
67 \( 1 + (-0.0871 + 0.996i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.573 - 0.819i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.828427326930427479006424136007, −17.00543306971674599045724396202, −16.57560069693415176792610720247, −16.12831522722247616867417217937, −15.01787277516048187811785675777, −14.79765578070715915846964102397, −14.01491166064999518233222537069, −13.13037819349745611833929505788, −12.333731654240069761790591792373, −12.226942775250640889733578619540, −11.35326401457454684038516954953, −10.65097416882900223660475704387, −9.72079798685039664653048242916, −9.13912436463282239940353417930, −8.70428432231196201488195055598, −7.630226105712742206607034180560, −7.31937207925934998745967236003, −6.44899556264353190468943771652, −5.55225004328472304095555285625, −4.8027590150883855137499041297, −4.29514103456911707770879854345, −3.54404502578398070264666634292, −2.660014517661825605845552652904, −1.56047592659707528223479884815, −1.06599077783811207067250143341, 0.34180167801973182944278891966, 1.26475386555532435784518815193, 2.45631037583242260652362022971, 3.083633685280340989081290455992, 3.67205255994852585804988590262, 4.48195094794091846119331890943, 5.42500800833816875489411222178, 6.003919296891151520458204042691, 6.89288087796068159965027634926, 7.587304899559432826693683901430, 7.844384053339359821417647087773, 8.967915280155747755409970975233, 9.68623102816167565532251620124, 10.10794984305732818118393246125, 11.26923625273074852446349167658, 11.43938896256001800764074874545, 12.081000524220863222278898088607, 12.97105110458892101924756596094, 13.73007187635925802277797866317, 14.407777445738754525958807476730, 14.8091735832747284910607138814, 15.60961723921002559715856087324, 16.1171840173466483020658337986, 16.98074709940107692404138473856, 17.46981556893802449831237758014

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