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Properties

Label 1-663-663.59-r0-0-0
Degree $1$
Conductor $663$
Sign $0.159 + 0.987i$
Analytic cond. $3.07895$
Root an. cond. $3.07895$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.707 − 0.707i)5-s + (0.965 − 0.258i)7-s − 8-s + (0.965 + 0.258i)10-s + (−0.258 + 0.965i)11-s + (0.707 + 0.707i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)19-s + (0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (0.965 + 0.258i)23-s − i·25-s + (−0.258 + 0.965i)28-s + (−0.258 + 0.965i)29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.707 − 0.707i)5-s + (0.965 − 0.258i)7-s − 8-s + (0.965 + 0.258i)10-s + (−0.258 + 0.965i)11-s + (0.707 + 0.707i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)19-s + (0.258 + 0.965i)20-s + (−0.965 + 0.258i)22-s + (0.965 + 0.258i)23-s − i·25-s + (−0.258 + 0.965i)28-s + (−0.258 + 0.965i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(3.07895\)
Root analytic conductor: \(3.07895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (0:\ ),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.614726716 + 1.374964260i\)
\(L(\frac12)\) \(\approx\) \(1.614726716 + 1.374964260i\)
\(L(1)\) \(\approx\) \(1.379141579 + 0.7067747617i\)
\(L(1)\) \(\approx\) \(1.379141579 + 0.7067747617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
11 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.258 - 0.965i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.258 - 0.965i)T \)
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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

−22.42473108032604540408606430233, −21.5491282543819216119610841298, −21.26040137166338048871806480679, −20.43639473237030465620803741743, −19.12570845494142933965537960893, −18.82834082423547206752912111439, −17.77789088743816459632248307320, −17.25892713967127572706395877621, −15.6931397755682279751936192479, −14.86630698423487611856306597279, −14.15982926159548276809038696350, −13.50297260776986021697594139977, −12.65257349101388140798086390512, −11.34577934543855297587027868452, −11.08935470023426713031165350290, −10.19216935531647837943932846136, −9.15212618754761955692940557208, −8.3435483032548992975855546839, −6.91561081239645537485963328438, −5.88075954825024397904505823895, −5.18989283052724688680293128113, −4.12388844195481477240846657396, −2.8557343845780104418955004372, −2.28524708020240617670861453063, −1.02306675332624555612910305173, 1.343677218207535762513513028, 2.5397568925326345650595552365, 4.093435650391084839326485986191, 4.80908606528658194152344166999, 5.502295436116763903566513061062, 6.53551019351215562995510604574, 7.58145205316259571802431770985, 8.29078139542913368844710022011, 9.21925259455437974678929335471, 10.13727965946537336839783883414, 11.40385592383259861697121221413, 12.46521192550091938541501844589, 13.040087522712029024343594945202, 13.93800979390561854666026188267, 14.72151975982223374789572410081, 15.36307346433524177024440800606, 16.6167889829395323437379464326, 17.0217338697536783383387525054, 17.84959030081892581186808594995, 18.46398046344501086598983525184, 20.03496468993522318428054309442, 20.88842212921416056394170768025, 21.24029263559577197078707883808, 22.24683958714724802867167405602, 23.25008941624432158131939930806

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