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Properties

Label 1-6048-6048.2131-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.735 + 0.677i$
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  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (0.906 + 0.422i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.422 + 0.906i)29-s + (0.173 − 0.984i)31-s + (−0.965 − 0.258i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (−0.173 − 0.984i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (0.906 + 0.422i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (0.422 + 0.906i)29-s + (0.173 − 0.984i)31-s + (−0.965 − 0.258i)37-s + (0.342 + 0.939i)41-s + (0.819 + 0.573i)43-s + (−0.173 − 0.984i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541359506 + 0.6021743496i\)
\(L(\frac12)\) \(\approx\) \(1.541359506 + 0.6021743496i\)
\(L(1)\) \(\approx\) \(1.029717497 + 0.08332025555i\)
\(L(1)\) \(\approx\) \(1.029717497 + 0.08332025555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.996 + 0.0871i)T \)
11 \( 1 + (0.0871 - 0.996i)T \)
13 \( 1 + (0.906 + 0.422i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.422 + 0.906i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (0.819 + 0.573i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.422 - 0.906i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−17.59391160895281980150173737883, −17.01400718159324309786973578987, −16.11753700236096782487577161394, −15.65281521976115573946723042284, −15.21784739452517651290047168478, −14.33480375776055245294667916706, −13.813707714748633179238286572067, −12.81884774988113181752559243107, −12.31127163034667316087448530672, −11.85757328988824447150126644873, −10.93263011484679155876301957819, −10.501072354401414831677834141177, −9.624959879913326287169798188809, −8.87784923537920870881519090794, −8.21693786397456708455770928161, −7.56845117798937415113716386256, −6.975548800649794231659471059209, −6.227928757199795522704134306938, −5.20425436763007607720083000469, −4.70769994998797930856786358298, −3.844279742626469899064787382992, −3.26292186725336546795948975131, −2.43919355642300503871129912526, −1.30857478351187551943775663368, −0.57551304262808022712041976459, 0.89698461699042952505943727309, 1.4053991732783820025894036138, 2.784248620124627799898115731686, 3.49074656386095511513627270521, 3.821964866303772520810061186121, 4.81189823514788548308744835646, 5.69366527431807329417773562976, 6.1727950662064945768607829899, 7.26139057432077985560300132356, 7.63077898371162243585873088963, 8.50237892181157847311272292865, 8.89930071219502126879405470333, 9.8645528193837806382677613924, 10.616045550350882199285695532007, 11.31255333732019504844878031439, 11.74944476763576502917523615318, 12.366808549238536838382787741846, 13.27515309468497667488436601715, 13.87740261833191175655181645774, 14.50338581009686952924278124207, 15.21753972806928260432983701272, 15.94028117101368651346806542943, 16.43689114677445682656538565887, 16.81765253792391036540458663996, 18.021089760292000175342368508064

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