↑

Properties

Label 1-6048-6048.2795-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

Related objects

Downloads

Learn more

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

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

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: $\chi_{6048} (2795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.816864033 - 0.04389517675i\)
\(L(\frac12)\) \(\approx\) \(2.816864033 - 0.04389517675i\)
\(L(1)\) \(\approx\) \(1.481543593 + 0.02267777544i\)
\(L(1)\) \(\approx\) \(1.481543593 + 0.02267777544i\)

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 \( 1 + (0.173 - 0.984i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.88275014568092478043991776494, −16.807584888699948378642049396, −16.42402042822576956559843899722, −16.11600756461322584180217366953, −14.847542493047143420165763742881, −14.37106054387605682543416069757, −13.72841918567944739785538490386, −13.342059038313667592037279637504, −12.39162935335278997807990348843, −11.88138112307394162298675444886, −10.950291877878024277680174059863, −10.418907400694517420109641678692, −9.78243566425965663942426348249, −8.99416726232554230677299316068, −8.46359984221257882966120570718, −7.77880311828846906963761372939, −6.62481281225002422476138654009, −6.30222590909675862316624362419, −5.480409597414580349732947301154, −5.00606967672754882404982358224, −3.83094468642712687664622405212, −3.26538680565980312015039660919, −2.44961579106921918014561062656, −1.40416624566030125630791158065, −0.98956722364179342140452780539, 0.860426157220604866001373614994, 1.57971095205389505655701787361, 2.40351913881042445488830659291, 3.12435295005647896408267861437, 4.00907022860667176767391010447, 4.81800513715817227951145699557, 5.71659875013121636637542770794, 5.95782613756599360572476077666, 6.97587603467711803744849787042, 7.57937130894384873184538492996, 8.31709130505413711935377110977, 9.300320887208708191873794541, 9.71066279629078254596808514277, 10.22657787058939207114192312879, 11.11298057214897591931376945517, 11.7264121707728465317065847152, 12.54750052376416030891099206849, 13.23118632948073932174009695601, 13.66906241664660214183361285642, 14.392669729111540388682918078155, 15.08605695294952976116441300372, 15.68364515459063839254693008135, 16.4616082399505024352421606397, 17.14699323451735329903241457497, 17.84061835510291266662429844050

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