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Properties

Label 1-663-663.608-r1-0-0
Degree $1$
Conductor $663$
Sign $0.362 - 0.931i$
Analytic cond. $71.2492$
Root an. cond. $71.2492$
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 + i·5-s + (0.866 − 0.5i)7-s − 8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s − 25-s + (−0.866 − 0.5i)28-s + (0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + i·5-s + (0.866 − 0.5i)7-s − 8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s − i·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s − 25-s + (−0.866 − 0.5i)28-s + (0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.362 - 0.931i$
Analytic conductor: \(71.2492\)
Root analytic conductor: \(71.2492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (1:\ ),\ 0.362 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.420620280 - 1.654847536i\)
\(L(\frac12)\) \(\approx\) \(2.420620280 - 1.654847536i\)
\(L(1)\) \(\approx\) \(1.379445378 - 0.6060115141i\)
\(L(1)\) \(\approx\) \(1.379445378 - 0.6060115141i\)

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 \( 1 + iT \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - iT \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−22.96757218228549876728933627738, −21.77358214680044611062468857205, −21.41645863496932924330967687190, −20.54352877755296588002757603893, −19.535791111739476844120527213524, −18.42210648127903268482959699830, −17.49220402622596540462410908122, −16.97817060192863177760704356888, −16.06007354141209237695379851992, −15.42806708281435616292868700867, −14.24268593470961338770093014527, −13.97246141514265900681693158316, −12.64671137055353763115486702003, −12.14609219602448047756167656930, −11.30515807399609195685397361335, −9.75345694849717479057209861036, −8.68963975895973956011453488645, −8.36877467805334878002807858269, −7.34073122721548774906775796713, −6.023933754322187198547382649305, −5.52763937990494711304880332013, −4.43204770169860709226922230978, −3.80912577556252977733093642282, −2.204495432770148121540550857582, −0.882952484046186321109366026055, 0.77722123082956930521799281942, 1.96754714716535473912435850982, 2.799576987493556886717406953674, 4.08472178768800240232993742402, 4.53645877992298932543961005453, 5.94871483789145361136110904110, 6.75833671890254287100044323968, 7.843572764376623833099086451538, 9.065656581237288142201151149113, 10.03084949698080335480106477345, 10.78558148281365141544408331958, 11.448590003116443788918380615195, 12.19224726235148159986464420527, 13.35766037726118283760711080241, 14.15044513254755385697798312605, 14.71125154407518906090243817388, 15.38275027062440608559579507116, 16.877720404739521489604981084515, 17.898952026179102864729214807577, 18.25915945754633632165598381455, 19.50094378313547956641300649742, 19.89373956723189469848076814973, 20.89341438277928457937336058189, 21.69132206781608444318798087699, 22.33582758634889918483101794541

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