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Properties

Label 1-648-648.277-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  + (0.993 − 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

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

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: $\chi_{648} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.605277697 - 0.6558281062i\)
\(L(\frac12)\) \(\approx\) \(1.605277697 - 0.6558281062i\)
\(L(1)\) \(\approx\) \(1.290077178 - 0.1985251542i\)
\(L(1)\) \(\approx\) \(1.290077178 - 0.1985251542i\)

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

−23.11264746850300093369238579506, −21.86980747114193137601283345512, −21.457818887178936655739350212619, −20.55436347492467315965537938289, −19.89005694806366692243445620082, −18.661764377792321352024626969417, −17.988295623247074298326777194069, −17.09180129299990393128051751097, −16.80515358771008036408424461868, −15.20097246430277516519086973810, −14.631139833034455788496034078565, −13.93560150455905657759762987866, −12.86850185299096923355350393110, −12.2265691481887315791438945197, −10.92178697279447013708598501257, −10.282631593052222018812403282898, −9.51793142683886785461802588701, −8.40381539826456854874330576793, −7.42792104524099995256332150109, −6.64044644925079411615455887674, −5.34501044322467055529951208885, −4.82697649192047966265716950711, −3.51908989484290067834899479227, −2.10590975415831144333254464060, −1.53081977104589266338021656805, 0.89026273390280475116154766645, 2.30410101370774734481342345211, 2.876937615989675273379946904651, 4.694836908825223479049981519962, 5.22397463863637537824166147486, 6.15764931232600463445555961120, 7.28910395557574398233670948432, 8.294307309357975390927042185516, 9.15964837507480896148513233616, 9.942041406062743688098813180894, 11.04257402902499451336108057405, 11.6695446640074125097979892556, 12.91118696257908916041640351873, 13.51854119011060295448385379294, 14.5013329753770446472038955004, 15.08077408324678050082594530814, 16.318859164789453351196946964611, 17.03510028093208053452197042053, 17.919752759940740244054692475377, 18.433523185995529412662196669054, 19.4723716000531972782759192895, 20.5117494196271968677620238972, 21.276278119163546910703235194800, 21.74297657672727454844000997716, 22.60586766600787240473230471724

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