Properties

Label 1-648-648.115-r1-0-0
Degree $1$
Conductor $648$
Sign $0.713 - 0.700i$
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  + (0.993 − 0.116i)5-s + (−0.893 − 0.448i)7-s + (0.396 + 0.918i)11-s + (−0.973 − 0.230i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.893 + 0.448i)23-s + (0.973 − 0.230i)25-s + (0.286 − 0.957i)29-s + (0.835 + 0.549i)31-s + (−0.939 − 0.342i)35-s + (0.939 − 0.342i)37-s + (−0.686 + 0.727i)41-s + (0.597 − 0.802i)43-s + (0.835 − 0.549i)47-s + ⋯
L(s)  = 1  + (0.993 − 0.116i)5-s + (−0.893 − 0.448i)7-s + (0.396 + 0.918i)11-s + (−0.973 − 0.230i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.893 + 0.448i)23-s + (0.973 − 0.230i)25-s + (0.286 − 0.957i)29-s + (0.835 + 0.549i)31-s + (−0.939 − 0.342i)35-s + (0.939 − 0.342i)37-s + (−0.686 + 0.727i)41-s + (0.597 − 0.802i)43-s + (0.835 − 0.549i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897390675 - 0.7751693898i\)
\(L(\frac12)\) \(\approx\) \(1.897390675 - 0.7751693898i\)
\(L(1)\) \(\approx\) \(1.161999798 - 0.1171676097i\)
\(L(1)\) \(\approx\) \(1.161999798 - 0.1171676097i\)

Euler product

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

−22.36247466649856817567770724591, −21.96410717712885345274276577159, −21.4744938654879941048950778035, −20.23128104712106051219154671049, −19.39111043240032580461482936560, −18.7643960796067993159500181661, −17.76592777266415547462739939683, −16.96584617511122678084353602658, −16.30555814564392728702867320934, −15.24926428838395405887601666100, −14.3457898402292997059306510847, −13.59319907618476915470092258497, −12.75395736487500066939779380062, −11.96039849150690554772142902675, −10.79002552009894104139262121934, −9.93554324800249865549561911860, −9.235454263648262308999886839906, −8.40844150148592899762988466125, −7.01089759856895086887858184549, −6.21115573514896963275136262972, −5.583181332402720513142109278433, −4.31662627393465459548114675162, −3.014574781659409566943131690549, −2.31038085423529299857469956753, −0.90439943022794301302784715516, 0.58490460378471160085998378658, 1.91563928050019489963674294579, 2.85042802118810602560198267645, 4.09261850539228102374335260524, 5.13099981557351420385248771036, 6.108349493797735519667338916902, 6.950384318738011738580502016078, 7.82174230718765176565771531481, 9.24723975481470779224010213671, 9.906640270338067453638129906841, 10.24567156000890860555898957612, 11.87663561367660269769599407317, 12.44984605604711226108749265027, 13.475432798349050947614351720170, 14.07504946041265867900828259315, 15.00432178648385538194965980323, 16.09373819112424393420228315047, 16.85562061052846965965301987341, 17.554176343660444606321315216251, 18.32953620539278397272346509203, 19.410604004238547816760074593215, 20.14940519060596951965003476121, 20.844537384205571029252623363599, 21.87238423164142878099231547012, 22.568479057253295299861473611735

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