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Properties

Label 1-648-648.59-r0-0-0
Degree $1$
Conductor $648$
Sign $0.0774 - 0.996i$
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  + (0.973 − 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

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

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.0774 - 0.996i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.0774 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035933753 - 0.9585391223i\)
\(L(\frac12)\) \(\approx\) \(1.035933753 - 0.9585391223i\)
\(L(1)\) \(\approx\) \(1.091446643 - 0.3318003151i\)
\(L(1)\) \(\approx\) \(1.091446643 - 0.3318003151i\)

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

−22.8495835408772734644139715941, −22.18070708203843983995109246084, −21.526759322385209953856612212709, −20.810567881736324869101949836277, −19.57815994687443101773361640056, −19.050390992724711211100667209, −18.08198284467740444185976406792, −17.2751172238063798649340163848, −16.64764286897222589454038897465, −15.500953424368537308680547227242, −14.63500358554701201667490396508, −14.08368166400178257350314795384, −12.78363838796775803910638385098, −12.41010844127787318559385583276, −11.271310877580304433562003194763, −10.155373866987836921703665369111, −9.39076329975171228343686521437, −8.94565174172556504211358399292, −7.32735657183223292524563196253, −6.680576197251371026037075794485, −5.66472994175256061501255134063, −4.89972728258293394231067132528, −3.48081278135449246410643163947, −2.44330696650237941259868380511, −1.5941246977435892717906177652, 0.689139477137518677085423388229, 1.94959786327593270887244059059, 3.149827925159435046636241636010, 4.13802185675961459910000776867, 5.34750771862690839408475958403, 6.19005152567025106031685066484, 6.99601422497312673939979606369, 8.155934236647265508219326073754, 9.14387644486945866341314487525, 10.04270827662264178308320976070, 10.52828038338774611249908585774, 11.800256077960570403859167058440, 12.853410992398992867574510981247, 13.34531641853496602831651176719, 14.37869142045981117220810182340, 14.93809662647928464967132672035, 16.50259275213783523346519992131, 16.846147038578454099531882159102, 17.43060916996303509955225980570, 18.760746605268924276508679690363, 19.28967006634846442508905783791, 20.35680286138019690608018436233, 20.96460478879828615150306918259, 21.98506872450967342389139886441, 22.48708331619730400494947421241

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