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Properties

Label 1-1323-1323.853-r1-0-0
Degree $1$
Conductor $1323$
Sign $0.143 + 0.989i$
Analytic cond. $142.176$
Root an. cond. $142.176$
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  + (−0.661 + 0.749i)2-s + (−0.124 − 0.992i)4-s + (0.969 + 0.246i)5-s + (0.826 + 0.563i)8-s + (−0.826 + 0.563i)10-s + (0.980 + 0.198i)11-s + (−0.878 + 0.478i)13-s + (−0.969 + 0.246i)16-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (0.124 − 0.992i)20-s + (−0.797 + 0.603i)22-s + (−0.124 − 0.992i)23-s + (0.878 + 0.478i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.661 + 0.749i)2-s + (−0.124 − 0.992i)4-s + (0.969 + 0.246i)5-s + (0.826 + 0.563i)8-s + (−0.826 + 0.563i)10-s + (0.980 + 0.198i)11-s + (−0.878 + 0.478i)13-s + (−0.969 + 0.246i)16-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (0.124 − 0.992i)20-s + (−0.797 + 0.603i)22-s + (−0.124 − 0.992i)23-s + (0.878 + 0.478i)25-s + (0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.143 + 0.989i$
Analytic conductor: \(142.176\)
Root analytic conductor: \(142.176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (1:\ ),\ 0.143 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.442593300 + 1.248924143i\)
\(L(\frac12)\) \(\approx\) \(1.442593300 + 1.248924143i\)
\(L(1)\) \(\approx\) \(0.9070982961 + 0.3722729371i\)
\(L(1)\) \(\approx\) \(0.9070982961 + 0.3722729371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.661 + 0.749i)T \)
5 \( 1 + (0.969 + 0.246i)T \)
11 \( 1 + (0.980 + 0.198i)T \)
13 \( 1 + (-0.878 + 0.478i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.124 - 0.992i)T \)
29 \( 1 + (0.921 - 0.388i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (-0.270 - 0.962i)T \)
43 \( 1 + (-0.969 + 0.246i)T \)
47 \( 1 + (0.318 + 0.947i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (0.411 + 0.911i)T \)
61 \( 1 + (0.124 - 0.992i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.733 - 0.680i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.878 - 0.478i)T \)
89 \( 1 + (0.988 + 0.149i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−20.352661514076674941529558583817, −19.80781833111743535545202982178, −19.30740221945842414517684286441, −18.02856766196033370832776010836, −17.69052856101331247770625678036, −17.07892888466739941026531617076, −16.30674463269168753356177523759, −15.2965456995259661941684608392, −14.22858752939515850926169591657, −13.43093243727181777704395198074, −12.88179107234840982869006483218, −11.87149454041227587307681905877, −11.32290138043366357856002053932, −10.20678330956410687785619311218, −9.74107552222936005284453719269, −8.953449753109471938894662714861, −8.3084064445705267585990575698, −7.10722331484356386001427425116, −6.45503087826110433156790872575, −5.193865704831042335694402259977, −4.39801665646035524428902090338, −3.208940336357551295068499150006, −2.402735173424677465897257692887, −1.48090377136801448995034856665, −0.599446701625115350676078953, 0.821583914847270930814765598892, 1.84035121593476744216123951241, 2.64306486481248565424354970002, 4.303496168838813025899959997837, 4.986269802178288467904576645081, 6.1944336831238767211218189671, 6.505715665496951489732215767936, 7.39264148117746345030127114964, 8.4167906602769477588313440825, 9.271342644776876988913131240822, 9.78645422027566925760035532878, 10.49153709165460462507166335262, 11.50807277582265297003023825880, 12.40892892242178800543318021127, 13.660992601502801865719684624808, 14.12412051905041720842477588358, 14.77891456396694592220681281921, 15.60788563500521854437695146550, 16.675796087084971435314667580335, 17.04215410517119308895150431227, 17.803579695930156881428165716, 18.46319678056078443850578008685, 19.23728871781940293713486289152, 20.02667915585685342980721458638, 20.769569487534819338313172246368

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