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Properties

Label 1-648-648.493-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

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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: $\chi_{648} (493, \cdot )$
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 \( 1 + (0.396 + 0.918i)T \)
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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

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

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