↑

Properties

Label 1-6048-6048.3643-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.419 - 0.907i$
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  + (−0.0871 − 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.422 − 0.906i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (0.173 − 0.984i)31-s + (0.258 − 0.965i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 − 0.996i)5-s + (0.996 + 0.0871i)11-s + (0.422 − 0.906i)13-s + 17-s + (0.707 − 0.707i)19-s + (−0.342 − 0.939i)23-s + (−0.984 + 0.173i)25-s + (0.906 − 0.422i)29-s + (0.173 − 0.984i)31-s + (0.258 − 0.965i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (−0.173 − 0.984i)47-s + (−0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110662384 - 1.737398113i\)
\(L(\frac12)\) \(\approx\) \(1.110662384 - 1.737398113i\)
\(L(1)\) \(\approx\) \(1.123746930 - 0.4546363082i\)
\(L(1)\) \(\approx\) \(1.123746930 - 0.4546363082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0871 - 0.996i)T \)
11 \( 1 + (0.996 + 0.0871i)T \)
13 \( 1 + (0.422 - 0.906i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.573 + 0.819i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.906 - 0.422i)T \)
61 \( 1 + (0.819 + 0.573i)T \)
67 \( 1 + (0.996 - 0.0871i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.906 + 0.422i)T \)
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

−17.92510626592655212739642739153, −17.34032881642983851435044907713, −16.4850859152944916229540978225, −16.057711293736052799310349092114, −15.21454121386704142809094241695, −14.50909372247119917175135682731, −14.021732929712983488608805059707, −13.67414286513564687578233175687, −12.45404680457530762256794575622, −11.836962293347236789180137409863, −11.451925715360354373089018825, −10.66396319571269057193715196152, −9.78776070417328473920600882763, −9.564933227423521399133488113969, −8.42689724234827831180285345061, −7.91778542412647518387388936052, −6.936008894197466885281934993375, −6.6113998113295707562298382807, −5.85568147161659564019097645270, −5.02752401629970307005606630965, −4.022584581256815402883433731009, −3.44970700074178720690003190421, −2.907922828945274622887615928177, −1.6688486225787858991616779364, −1.247665842161542227459065208559, 0.59938387018273622360312708436, 1.0900586699403424275964897728, 2.08674127009111846270718965375, 3.057660325496404648599196986773, 3.85057577123320910055794114536, 4.48661469905814608372467907165, 5.29243273223601092656973614400, 5.88119806496656893299661621304, 6.64323958999096054115303021036, 7.58114738423739664746062065629, 8.17172071777070738887155437167, 8.78731479716199406106857003571, 9.51478235184462383778446764332, 10.047306639581151522850550913258, 10.922138463245748674123554995150, 11.778769577335120031057549232747, 12.16168529411388859689483440002, 12.86647089755947941364928304284, 13.51582776707441154326944891957, 14.186582235846660881554046683478, 14.88422585091566079131535474039, 15.70334731417249070236939348263, 16.1349505314125962964664943753, 16.954712622594584694064086739991, 17.29722124492738365854260547897

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