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Properties

Label 1-432-432.365-r1-0-0
Degree $1$
Conductor $432$
Sign $-0.969 + 0.244i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.969 + 0.244i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ -0.969 + 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008357873702 - 0.06726052953i\)
\(L(\frac12)\) \(\approx\) \(0.008357873702 - 0.06726052953i\)
\(L(1)\) \(\approx\) \(0.7475165065 - 0.06769399175i\)
\(L(1)\) \(\approx\) \(0.7475165065 - 0.06769399175i\)

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

−24.51155448778070222722897871493, −23.13242744920934986431703588406, −23.03545187813098229934254838337, −21.775239439650932697630304555661, −21.01994300564221009318015697174, −19.87259943504859115357149824663, −19.077516175403063788785710548388, −18.77676369818252589842353103031, −17.26945029895299062723863449944, −16.40904328180461961136830636322, −15.76953808819179222938063397560, −14.85447926833697204303333226386, −13.89178867142532039537135863434, −12.75864420062602602472681482532, −11.85416418224762727215627786618, −11.42049295801676122932693394814, −9.95569726746009310521667514131, −9.029776802348713496245488053474, −8.313429077209620785603938731939, −6.9955876725763230694595601373, −6.31362264185757746919112542758, −4.95269193862603012465996444411, −3.86868649985324358074679636879, −3.03418268055353193596251005097, −1.44934117789728740360004482928, 0.02054489838811736111138775431, 1.20198373902925538773809383156, 3.06161156310112657513663974845, 3.7552391805022762818127847962, 4.76297064084802484275875676399, 6.33687057464810923041772253536, 6.9495477631870292237096862375, 8.08915498276150996187487023538, 8.9118842407923518320267426545, 10.26623327197655751995877345149, 10.82273081956896270443921983476, 12.04456290464148508771138816333, 12.72976374613978624392208870367, 13.724227058070276020790877640502, 14.957430590103664127437162060644, 15.39039085553176796553686763177, 16.6616124651918077398553524003, 17.0843056007548488554012437778, 18.42306837020295595858729762580, 19.36776260676507693732909046449, 19.81019601899169784505682309868, 20.67710192767833323663264323724, 21.94385489417549586268979327127, 22.773042051141161357197608022407, 23.27653066305117394113726130513

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