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Properties

Label 1-432-432.149-r1-0-0
Degree $1$
Conductor $432$
Sign $0.512 - 0.858i$
Analytic cond. $46.4248$
Root an. cond. $46.4248$
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  + (0.984 − 0.173i)5-s + (−0.766 − 0.642i)7-s + (−0.984 − 0.173i)11-s + (−0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.342 − 0.939i)29-s + (0.766 − 0.642i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.766 − 0.642i)47-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (−0.766 − 0.642i)7-s + (−0.984 − 0.173i)11-s + (−0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.342 − 0.939i)29-s + (0.766 − 0.642i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.766 − 0.642i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.689864787 - 0.9593267205i\)
\(L(\frac12)\) \(\approx\) \(1.689864787 - 0.9593267205i\)
\(L(1)\) \(\approx\) \(1.148109341 - 0.1780628534i\)
\(L(1)\) \(\approx\) \(1.148109341 - 0.1780628534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−24.160860266699036134590268172614, −22.9703245502299383089506157092, −22.35094486510013447266838362396, −21.52854210922082984937391315772, −20.709184053184520007645399480721, −19.80678511570356720947396597861, −18.62397628349390207376517498492, −18.12114684886925051553747155457, −17.23264941204693475043358920293, −16.10304606921118825320069355700, −15.42083911447394685782516112478, −14.39067314988561045959215732788, −13.295145178979236070654317438190, −12.84229458316034055281276067607, −11.694003360068122174768885937670, −10.45026431677069730009733431879, −9.76771861896157106239964999221, −8.9997813328073699279953030042, −7.66853639351099754995333436472, −6.73038534984726995098307600843, −5.50115897422826622220560748473, −5.1248972261592290718414751540, −3.01746441568446428253030029895, −2.7063171238272909299884892190, −1.04085337806820626664286046233, 0.592205038118784102392454903404, 1.96422858401266156909571560737, 3.07374721238048358228900238713, 4.32695204851327794761923999472, 5.50304004584492733031508856011, 6.34537262363341341963153363298, 7.35376561990206127374003788071, 8.488851187751005012254819459860, 9.75235013416018232993703863945, 10.062066603853273632580573663066, 11.222370312420831786875187869972, 12.52563969196741440729850915132, 13.24358885889608979242086294222, 13.945870137012156244396675455541, 14.91886480142531486315698609977, 16.226191277741564955622677652433, 16.73663252145583360480377420370, 17.60045965811744669825039499231, 18.69602193055568690250832388481, 19.30856414873502739590318227815, 20.61014523216593742623316005364, 21.057647076788618480698069680136, 22.04494415278663474267143727052, 22.88704374786964104043046793772, 23.80486576868753424716012051138

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