Properties

Label 1-648-648.227-r0-0-0
Degree $1$
Conductor $648$
Sign $-0.902 - 0.431i$
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  + (−0.286 − 0.957i)5-s + (−0.396 − 0.918i)7-s + (−0.973 + 0.230i)11-s + (0.835 + 0.549i)13-s + (0.939 − 0.342i)17-s + (−0.939 − 0.342i)19-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + (−0.0581 − 0.998i)29-s + (0.993 − 0.116i)31-s + (−0.766 + 0.642i)35-s + (−0.766 − 0.642i)37-s + (−0.893 + 0.448i)41-s + (−0.686 − 0.727i)43-s + (−0.993 − 0.116i)47-s + ⋯
L(s)  = 1  + (−0.286 − 0.957i)5-s + (−0.396 − 0.918i)7-s + (−0.973 + 0.230i)11-s + (0.835 + 0.549i)13-s + (0.939 − 0.342i)17-s + (−0.939 − 0.342i)19-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + (−0.0581 − 0.998i)29-s + (0.993 − 0.116i)31-s + (−0.766 + 0.642i)35-s + (−0.766 − 0.642i)37-s + (−0.893 + 0.448i)41-s + (−0.686 − 0.727i)43-s + (−0.993 − 0.116i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ -0.902 - 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1593377455 - 0.7025889179i\)
\(L(\frac12)\) \(\approx\) \(0.1593377455 - 0.7025889179i\)
\(L(1)\) \(\approx\) \(0.7580005052 - 0.3202262253i\)
\(L(1)\) \(\approx\) \(0.7580005052 - 0.3202262253i\)

Euler product

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

−23.180928574658283968946499643645, −22.5080554077576589063775417884, −21.45784809576310139175653140698, −21.08315486540433658518861104068, −19.75846429311266660928320312450, −18.87584497165261792686940933064, −18.53909226985910870833699570486, −17.65795832405565048224576849782, −16.47380335667055113579546803717, −15.491398677299458704283394069970, −15.19960850438624218250265834050, −14.10943054563367033337610965751, −13.1318250685825856475761122704, −12.34318694244449480747815946964, −11.34850649318441656676261404167, −10.537980448147819768445855329431, −9.83158316017141553734101249921, −8.48544318511165063233373987299, −7.9473522587217082771439291250, −6.70332028730232174167470902725, −5.946202666893995592533142766042, −5.05317892479790997829391428339, −3.38809777699516619012256082829, −3.069296275100332459426409331510, −1.71890352064407776053136530935, 0.35149120608101626813562859709, 1.57997977714717758428303016154, 3.0254748641534615320305644977, 4.16339224868999576106259142655, 4.81725689611807792905730854740, 6.002028703638121271920063132496, 7.03608249553259968248936151378, 8.010593429835533350684497098833, 8.72347038100720618267941342147, 9.84210506227019975594677975645, 10.57043965243827784992758813006, 11.60705722452505563862701810796, 12.58253539210789606725403137080, 13.27709273725690309321165436637, 13.93905099411429748733175497038, 15.23030341816903489607940055503, 16.004002396792563830557828073960, 16.71924401298723031418816778874, 17.32303078043913960359265823356, 18.57992821282617288968268931746, 19.22457152737406277672240221289, 20.225777341599966547255792243254, 20.85652990135770750032907906343, 21.364441601367556679935682889233, 22.974700506585982893713729269514

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