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Properties

Label 1-432-432.293-r1-0-0
Degree $1$
Conductor $432$
Sign $-0.999 + 0.0145i$
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  + (−0.342 + 0.939i)5-s + (−0.173 + 0.984i)7-s + (0.342 + 0.939i)11-s + (−0.642 − 0.766i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.642 + 0.766i)29-s + (0.173 + 0.984i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.173 + 0.984i)47-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (−0.173 + 0.984i)7-s + (0.342 + 0.939i)11-s + (−0.642 − 0.766i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.642 + 0.766i)29-s + (0.173 + 0.984i)31-s + (−0.866 − 0.5i)35-s + (0.866 − 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.173 + 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.999 + 0.0145i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ -0.999 + 0.0145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008322708376 + 1.144434378i\)
\(L(\frac12)\) \(\approx\) \(0.008322708376 + 1.144434378i\)
\(L(1)\) \(\approx\) \(0.8193535601 + 0.4379966177i\)
\(L(1)\) \(\approx\) \(0.8193535601 + 0.4379966177i\)

Euler product

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

−23.55769171281038995772729662611, −22.73541310975017736460372347162, −21.72416224811784932401414216505, −20.73971665559730546901744344390, −20.09401122235584094469173345119, −19.31053000823583557799692057676, −18.39974362466742675674010925384, −16.95721941622488371942792045291, −16.69922011129735134694110857027, −15.89672673745004748388489137852, −14.58233140145228868806637176278, −13.71098595979879409552923439073, −13.019511889600551315500549022950, −11.79312876973038699088896149746, −11.27636269640470435587442222512, −9.834137757403387322012490574410, −9.20725242803146047600694516110, −8.02437155750789891245774955464, −7.246640951855952821208566019915, −6.06524157007989340552035265004, −4.81082791860655366403725109188, −4.09858925948278877962143315379, −2.86650778015394311083002132922, −1.16426060759691485735740801812, −0.3352609007288726730021059775, 1.66803540326502751199141964732, 2.84873702866061542692608528985, 3.700321445871199235184168121960, 5.17041273768860769221491202347, 6.0506230683752553677063441319, 7.22296926432303540152582955547, 7.89557072240606297675542760754, 9.245414960198286053024297636322, 10.0394477167027005619255054251, 11.00504473373716831493165902854, 12.14147974201584503973738409828, 12.552760255488878263219502247998, 14.02084532031850558018807966824, 14.98168278658264631432802421320, 15.27923849149266437301635848198, 16.4421276345374286548889269584, 17.72625969087957508054822305531, 18.16390029920831045809421154677, 19.28729563908181179240226116433, 19.77511890454709510327260775619, 21.01473502585655957503045790754, 22.010347385443992489196013488215, 22.51349290831582726656600647604, 23.31320112033138803458664596710, 24.39965168121095500429514752525

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