Properties

Label 1-648-648.83-r0-0-0
Degree $1$
Conductor $648$
Sign $0.268 - 0.963i$
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  + (0.893 − 0.448i)5-s + (0.286 − 0.957i)7-s + (0.0581 + 0.998i)11-s + (−0.597 − 0.802i)13-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 − 0.957i)23-s + (0.597 − 0.802i)25-s + (0.396 + 0.918i)29-s + (0.686 − 0.727i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.993 − 0.116i)41-s + (−0.835 + 0.549i)43-s + (−0.686 − 0.727i)47-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)5-s + (0.286 − 0.957i)7-s + (0.0581 + 0.998i)11-s + (−0.597 − 0.802i)13-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 − 0.957i)23-s + (0.597 − 0.802i)25-s + (0.396 + 0.918i)29-s + (0.686 − 0.727i)31-s + (−0.173 − 0.984i)35-s + (−0.173 + 0.984i)37-s + (0.993 − 0.116i)41-s + (−0.835 + 0.549i)43-s + (−0.686 − 0.727i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.268 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256040364 - 0.9541545820i\)
\(L(\frac12)\) \(\approx\) \(1.256040364 - 0.9541545820i\)
\(L(1)\) \(\approx\) \(1.175013472 - 0.3397731731i\)
\(L(1)\) \(\approx\) \(1.175013472 - 0.3397731731i\)

Euler product

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

−22.8756634314238615476289631637, −21.95029831108019556319331208520, −21.54875159745139235679743001037, −20.92152309627604490781705502368, −19.46695565449547208572860409229, −19.0163533463928111709741873428, −17.979012953404752472252716527320, −17.50135012753595993255227687707, −16.40894937234621913942257627025, −15.577865411800484467980032748549, −14.55309805098974954187316248477, −13.98411701703711152811408020019, −13.12213373484307931036847272383, −11.96894290553868164181314066291, −11.331108187161330747610099686730, −10.27810303761846758660183393601, −9.38170024331794772024439101572, −8.6796454622859661195339204387, −7.58486002569695867357378091606, −6.338110448455882382088239037375, −5.83204923784911839404399514410, −4.817326984642782105147791861976, −3.44110193671024589810716876475, −2.40976586465582544207613357382, −1.566914960947728304877282206409, 0.790258039251495290305419205241, 1.98425193055152399245680420861, 3.02111947033041667628624133102, 4.71601518948031716253220830897, 4.82730775286175362390185657814, 6.321415963264534420390363539810, 7.14846756854462946098414861707, 8.064800166931095578388176293582, 9.234350668426918741685487473122, 9.96919306306157291292418587721, 10.648210505571556798560388206487, 11.831915506919055163332229137820, 12.79123534842928621142109888305, 13.48595210374226784775799588676, 14.25744457770694486608872355465, 15.15702772722061441883793756118, 16.209938475511362176166192679219, 17.08560464244750253561407959680, 17.7211904419900684812059983701, 18.240601341631702413440989993515, 19.82675880888545506315919641226, 20.25872813366568345112218845356, 20.831330829983103812376844269599, 22.038029419726639395346746093577, 22.55907798749923505685218374237

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