↑

Properties

Label 1-648-648.301-r0-0-0
Degree $1$
Conductor $648$
Sign $0.713 + 0.700i$
Analytic cond. $3.00929$
Root an. cond. $3.00929$
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  + (−0.396 − 0.918i)5-s + (−0.0581 + 0.998i)7-s + (−0.597 − 0.802i)11-s + (0.686 + 0.727i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (−0.973 − 0.230i)29-s + (0.893 + 0.448i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.286 + 0.957i)41-s + (0.993 − 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯
L(s)  = 1  + (−0.396 − 0.918i)5-s + (−0.0581 + 0.998i)7-s + (−0.597 − 0.802i)11-s + (0.686 + 0.727i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (−0.973 − 0.230i)29-s + (0.893 + 0.448i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.286 + 0.957i)41-s + (0.993 − 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.048023567 + 0.4281647422i\)
\(L(\frac12)\) \(\approx\) \(1.048023567 + 0.4281647422i\)
\(L(1)\) \(\approx\) \(0.9631318563 + 0.07539585847i\)
\(L(1)\) \(\approx\) \(0.9631318563 + 0.07539585847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.396 - 0.918i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.597 - 0.802i)T \)
13 \( 1 + (0.686 + 0.727i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.0581 - 0.998i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (-0.286 + 0.957i)T \)
43 \( 1 + (0.993 - 0.116i)T \)
47 \( 1 + (0.893 - 0.448i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (0.835 + 0.549i)T \)
67 \( 1 + (-0.973 + 0.230i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.396 - 0.918i)T \)
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

−22.877046984462453537338122804, −22.19088650955278666573408767, −20.96797689462012619411202611577, −20.306303666872906002904907266055, −19.56066930430800210712406068374, −18.59167318058409668092116965808, −17.854491571699407677033526027400, −17.18000446052150205686546163219, −15.84492735905941824323170839721, −15.47147608007800556755692654660, −14.422753337372119469571191698879, −13.57700931831293108047393296853, −12.8831494171806821824552422262, −11.59364598655515378938977132424, −10.91743508326887261903661194423, −10.18987824108883318508807002015, −9.29401387273973708260487066861, −7.77125568097346171149896556487, −7.43659492656824378719124575845, −6.49630100752225915055416174160, −5.2830330662547480523926023233, −4.17435514267685520427428392797, −3.289987783331921956305807684330, −2.29739986427231944916960729707, −0.64213943151522388516960254837, 1.16875679389531347792042392600, 2.35503600301387758168782148298, 3.63001027900350507627604889742, 4.535612467546266792980936942611, 5.71223690565240739255785284508, 6.206404550562615141957506190178, 7.82759447891003130419469290362, 8.49507050131940557213171779875, 9.06762423230818142039656042971, 10.280169277856873431891893914927, 11.30150544497966427245668280816, 12.12795553311709006441447559061, 12.808221018810127058281119157350, 13.664817358083437799662723818665, 14.79368152402065774387767218174, 15.59760759477619814735834622835, 16.4154457195396195288396855863, 16.89739979591054040887483411312, 18.30691226273630979645438832725, 18.82096640873894681492100484978, 19.58239367730738983340734408009, 20.780503609692226463256352957148, 21.167876926662342972070482487900, 22.0206791873207699939540047679, 23.13217984555401483607715195067

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