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Properties

Label 1-648-648.347-r0-0-0
Degree $1$
Conductor $648$
Sign $0.700 - 0.713i$
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  + (0.396 + 0.918i)5-s + (0.0581 − 0.998i)7-s + (−0.597 − 0.802i)11-s + (0.686 + 0.727i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (0.973 + 0.230i)29-s + (−0.893 − 0.448i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (0.286 − 0.957i)41-s + (−0.993 + 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯
L(s)  = 1  + (0.396 + 0.918i)5-s + (0.0581 − 0.998i)7-s + (−0.597 − 0.802i)11-s + (0.686 + 0.727i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (0.973 + 0.230i)29-s + (−0.893 − 0.448i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (0.286 − 0.957i)41-s + (−0.993 + 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297208169 - 0.5447041728i\)
\(L(\frac12)\) \(\approx\) \(1.297208169 - 0.5447041728i\)
\(L(1)\) \(\approx\) \(1.115526922 - 0.1272637753i\)
\(L(1)\) \(\approx\) \(1.115526922 - 0.1272637753i\)

Euler product

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

−23.21750507104247627109361120291, −21.87551781326701309236062186750, −21.425472326273758434853907409001, −20.50191172068228612849427045179, −19.90589287324029813142787958059, −18.74689551819548022625006135988, −17.94701038883290716324614061154, −17.36995293858479874284854501489, −16.226235732118232217678317087020, −15.58204391256872476027752087541, −14.80354729628305221029846246232, −13.64631741169080231235812488602, −12.67118701265623026868012880678, −12.4240532204802005102246654427, −11.1901434397018315028870298283, −10.10428118340981965983680701889, −9.37038220940595361952626496950, −8.36631720639184255309329525057, −7.82927408454049778814588252670, −6.22488700568796583152523332748, −5.58647863154156127580260249418, −4.75731000035198704491646613078, −3.5342327108837149272317359821, −2.23325435243364405788038237653, −1.35263787323935037756751210637, 0.74357335486228251363179538752, 2.278839327452478547767995884944, 3.187687146470480229480817053417, 4.22367632420548447263956686771, 5.35546999200191494299246626403, 6.552712049428900628732485191475, 7.03233497965940506956203964992, 8.13470482535409552510020502730, 9.20519639930571533934672101773, 10.18533149157943412249566228521, 10.99318913615227811201401991434, 11.44217535004023344712487842841, 12.96501974772774636096346033344, 13.85259864379708065351686377339, 14.06967569389253439476994062155, 15.312671885165836958526225153480, 16.22351314365201531960728550038, 16.91854351001568004162185006202, 18.09690995947832439146146645561, 18.43937192432784532526399762318, 19.45342875013545975225625659919, 20.37334567951356402676043812759, 21.17379744838507109841727038017, 21.95641295348028780015324282878, 22.76283095539564897989228095661

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