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Properties

Label 1-648-648.427-r1-0-0
Degree $1$
Conductor $648$
Sign $0.996 + 0.0774i$
Analytic cond. $69.6372$
Root an. cond. $69.6372$
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  + (−0.973 + 0.230i)5-s + (−0.597 − 0.802i)7-s + (−0.686 + 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.597 + 0.802i)23-s + (0.893 − 0.448i)25-s + (0.835 + 0.549i)29-s + (−0.396 − 0.918i)31-s + (0.766 + 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.0581 − 0.998i)41-s + (−0.286 − 0.957i)43-s + (−0.396 + 0.918i)47-s + ⋯
L(s)  = 1  + (−0.973 + 0.230i)5-s + (−0.597 − 0.802i)7-s + (−0.686 + 0.727i)11-s + (−0.893 − 0.448i)13-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.597 + 0.802i)23-s + (0.893 − 0.448i)25-s + (0.835 + 0.549i)29-s + (−0.396 − 0.918i)31-s + (0.766 + 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.0581 − 0.998i)41-s + (−0.286 − 0.957i)43-s + (−0.396 + 0.918i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.996 + 0.0774i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ 0.996 + 0.0774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5114055843 + 0.01984486567i\)
\(L(\frac12)\) \(\approx\) \(0.5114055843 + 0.01984486567i\)
\(L(1)\) \(\approx\) \(0.6065328243 + 0.01593989123i\)
\(L(1)\) \(\approx\) \(0.6065328243 + 0.01593989123i\)

Euler product

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

−22.696614477460129503191591244392, −21.744594746744444870604760903131, −21.247818554040509901075436984848, −19.83630030535404403728237486136, −19.509863660480844682891677724466, −18.70360689287484502857029966284, −17.81918663022263655324928146767, −16.61331630692020161851203651164, −16.02691958056348743445660138710, −15.29218263511221123146506322772, −14.51288220645948745185004273050, −13.23304362751378200794387738603, −12.55298920124667495275459827722, −11.7910079791743422114793252878, −10.910661787171893235490520229952, −9.929936375448331625209549005735, −8.66425401025705138805459348278, −8.38936361026259283504109504590, −7.062437891464203160875021368636, −6.26658883279954250651988476572, −5.06484122096741927523241596048, −4.22220612816705321197267879980, −3.049932632546950725191749150061, −2.17060916953438441825053246419, −0.32023828031815510796728351011, 0.36472047721226779641622567204, 2.121738993832751913405532773789, 3.22614449305689144994963832702, 4.185321832825236511975805598810, 4.975592329013544467727022541832, 6.42034929875409226278463342309, 7.304060709596091753583627771943, 7.81695441449639361466589114251, 9.018312551386405814697210218661, 10.19940483454245907296129248900, 10.633137524268632025825252543192, 11.82302264044621258320644708169, 12.58311524118500372295661878374, 13.37402227497589998644521178678, 14.4341314255387738749238592504, 15.39611290351185614597619100866, 15.83200743842195608322348233707, 16.93668422773048826917996520056, 17.659102716134761275207126552472, 18.69329392423995611427035283537, 19.560422144478472928700698778118, 20.06345323655646780911483812878, 20.82594603339068488531997097891, 22.209758066065250238135791035051, 22.595819689983879822872295992470

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