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Properties

Label 1-6048-6048.4405-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.999 + 0.0311i$
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 + (−0.0871 + 0.996i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.642 − 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (−0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.573 − 0.819i)5-s + (0.819 − 0.573i)11-s + (−0.0871 + 0.996i)13-s − 17-s + (−0.707 + 0.707i)19-s + (−0.642 − 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (−0.939 − 0.342i)31-s + (0.965 − 0.258i)37-s + (−0.642 − 0.766i)41-s + (−0.906 − 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0311i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007610654253 - 0.4883948493i\)
\(L(\frac12)\) \(\approx\) \(0.007610654253 - 0.4883948493i\)
\(L(1)\) \(\approx\) \(0.9532474540 - 0.1967736733i\)
\(L(1)\) \(\approx\) \(0.9532474540 - 0.1967736733i\)

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 \( 1 + (-0.0871 + 0.996i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.996 - 0.0871i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.906 - 0.422i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.996 - 0.0871i)T \)
61 \( 1 + (-0.422 + 0.906i)T \)
67 \( 1 + (0.819 + 0.573i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.996 + 0.0871i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−18.0348660734898992571231849959, −17.34676904721280016091704373775, −17.06957705063442740305056153174, −15.87463969877383917952014018826, −15.35175043044435430533876133833, −14.77428069309847348769540935990, −14.18225064642368370671596082656, −13.41659267767416979149279313034, −12.90968372092957541417896057534, −12.113668014075514965345478748120, −11.26236208656343274710848296892, −10.8311465437865657151011634175, −10.021562099741308863270464140934, −9.52136261126279666263747790361, −8.78665112591355966718769878765, −7.92868775002273443192044826068, −7.19700939621960947668401164404, −6.47185282497310751546787049952, −6.12040711570252626445419385132, −5.0795660886676441233165031138, −4.44009652990174160579266557067, −3.49774941141618219384100685415, −2.823352343592451629727263092715, −2.04944599851506894666326252749, −1.331075343739951436715199959509, 0.11175513448602741286456036826, 1.27616923701178932035643520898, 1.93050521660354770829977539298, 2.625468974306721890318991539043, 4.010769639644896710593363647970, 4.17639097249027928049089389671, 5.09697578801356520914605805351, 6.00645877327326006307859832984, 6.403237810185348176785923616654, 7.15322655476713682529541140541, 8.3650294335606008509423239720, 8.61580007279624631708076775371, 9.33222569266015932539462539470, 9.94224557980077930283085214454, 10.792065741140999385337608405560, 11.4562800211210049349769131926, 12.26743402228632531945199702957, 12.6208002559177465696125475852, 13.66903163919810932563599561634, 13.90750493558109879208053854809, 14.64162536900381314768598751173, 15.46727726918792405615910900689, 16.325437228129939431314697827383, 16.70160629888769321633979944435, 17.19280028767742817640194970197

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