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Properties

Label 1-432-432.61-r0-0-0
Degree $1$
Conductor $432$
Sign $0.0145 + 0.999i$
Analytic cond. $2.00619$
Root an. cond. $2.00619$
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

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

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0145 + 0.999i$
Analytic conductor: \(2.00619\)
Root analytic conductor: \(2.00619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (0:\ ),\ 0.0145 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7928101502 + 0.7813622437i\)
\(L(\frac12)\) \(\approx\) \(0.7928101502 + 0.7813622437i\)
\(L(1)\) \(\approx\) \(0.9639202381 + 0.2854694852i\)
\(L(1)\) \(\approx\) \(0.9639202381 + 0.2854694852i\)

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

−24.171472485853383393429296068583, −23.008765957566311393833169829296, −22.042340015984249494338412135267, −21.429889934792033946950606140437, −20.450289361821588245026941910824, −19.759015522826036082769863244656, −18.574368753641461638240472961748, −17.98578532160650166873538817708, −16.84861884942329719110404363650, −16.073510662696040334230571367335, −15.389029055549409327327337815368, −14.12545803059003783516976739148, −13.319552830929981421190315891497, −12.35554699634398743061311334485, −11.767714407040001052251164089126, −10.410345257314010896241887793016, −9.46761029320035384054526297819, −8.63312670272467869584300378350, −7.87076457626185810138749568890, −6.36300283056078657889373765344, −5.44944206845933346751457437489, −4.79134070924590692776310373968, −3.1729408592196611458988969010, −2.227001960423105528575070793641, −0.63181574499973800375185392659, 1.60314581419302753041993325939, 2.73692125842586902220691657740, 3.90653711323779092140406769996, 4.91542461498894830938861963619, 6.31994131054863848716391463553, 7.09936818036079274931431558537, 7.779564815111475147305458372782, 9.43073829112986773463155564815, 10.02603417338391284243043828809, 10.91488783200345551377188685747, 11.83216287398582761591928139208, 13.09950241492459386791724430007, 13.79665873562066822242348097305, 14.70239136645622427936637288270, 15.468329568079290237698282266434, 16.64982394183802270364053322698, 17.53760227170348783524493095338, 18.10234759633002863835784109864, 19.32312506666557153070772398919, 19.89131666129287618403563950446, 20.94065601977845662074502789837, 21.92898731211695913379319143156, 22.528969167971613732551080699432, 23.528753211680323373399741714329, 24.10683441294255760871892443646

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