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Properties

Label 1-648-648.61-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  + (−0.597 − 0.802i)5-s + (−0.835 + 0.549i)7-s + (0.993 + 0.116i)11-s + (0.286 + 0.957i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (−0.835 − 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.686 + 0.727i)29-s + (−0.0581 + 0.998i)31-s + (0.939 + 0.342i)35-s + (0.939 − 0.342i)37-s + (0.973 + 0.230i)41-s + (−0.396 − 0.918i)43-s + (−0.0581 − 0.998i)47-s + ⋯
L(s)  = 1  + (−0.597 − 0.802i)5-s + (−0.835 + 0.549i)7-s + (0.993 + 0.116i)11-s + (0.286 + 0.957i)13-s + (0.173 − 0.984i)17-s + (−0.173 − 0.984i)19-s + (−0.835 − 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.686 + 0.727i)29-s + (−0.0581 + 0.998i)31-s + (0.939 + 0.342i)35-s + (0.939 − 0.342i)37-s + (0.973 + 0.230i)41-s + (−0.396 − 0.918i)43-s + (−0.0581 − 0.998i)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: $\chi_{648} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027183300 - 0.4196505562i\)
\(L(\frac12)\) \(\approx\) \(1.027183300 - 0.4196505562i\)
\(L(1)\) \(\approx\) \(0.9287547301 - 0.1298532989i\)
\(L(1)\) \(\approx\) \(0.9287547301 - 0.1298532989i\)

Euler product

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

−22.88247148224517302652050569532, −22.38088242632694295907367343218, −21.48709219285591481464950068190, −20.27806620449203352170068805047, −19.597799196154128092628660619590, −19.07852934608101372677109971497, −18.07872904945480348524467460898, −17.158228636376336860589492072539, −16.34065300974718029729683079068, −15.462737213884686209554003776915, −14.69678309133821920556780522162, −13.87822825013357403782274181953, −12.88042089743225154377454142392, −12.04287392537293086543916533493, −11.09674105757995926733323545374, −10.260406075473713601642819100870, −9.58780485345041733337562462121, −8.16057796980124545378471122469, −7.61273989805863802925988090353, −6.33247452358621045171750089528, −6.01654833316113518326126435251, −4.07716205961603935231805810984, −3.722938783403773683988494619331, −2.6167957407723428628255305885, −1.03033892098900547527411644242, 0.70086067639545844249471107662, 2.10548153499623266807707535393, 3.38547387388326367967828136179, 4.30095811098216252080705193950, 5.20639912863657162890786839288, 6.459412263515315505029116664833, 7.08380911214324205852347133612, 8.48057448346303815236517410608, 9.068840187563756483614739256811, 9.75326122718185719516319620046, 11.18352373854313569051926937317, 11.970771791681902910635516075056, 12.50067984705724781469602882655, 13.547858319373964475998380720749, 14.41302232974285882483873945113, 15.510082518543825851501000446205, 16.2487140557841853548173837294, 16.66643971454127172966382426049, 17.89254583856277252079664667980, 18.75763508228153047872981229823, 19.76775654907360983504703684571, 19.943045387061281122078394934700, 21.235829987657236489408541788222, 21.92201624815529312172063795342, 22.820002210694217392739743216374

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