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Properties

Label 1-1323-1323.1222-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.863 - 0.504i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.998 − 0.0498i)2-s + (0.995 + 0.0995i)4-s + (−0.661 − 0.749i)5-s + (−0.988 − 0.149i)8-s + (0.623 + 0.781i)10-s + (−0.998 − 0.0498i)11-s + (−0.797 + 0.603i)13-s + (0.980 + 0.198i)16-s + (−0.900 + 0.433i)17-s + 19-s + (−0.583 − 0.811i)20-s + (0.995 + 0.0995i)22-s + (−0.411 − 0.911i)23-s + (−0.124 + 0.992i)25-s + (0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (−0.998 − 0.0498i)2-s + (0.995 + 0.0995i)4-s + (−0.661 − 0.749i)5-s + (−0.988 − 0.149i)8-s + (0.623 + 0.781i)10-s + (−0.998 − 0.0498i)11-s + (−0.797 + 0.603i)13-s + (0.980 + 0.198i)16-s + (−0.900 + 0.433i)17-s + 19-s + (−0.583 − 0.811i)20-s + (0.995 + 0.0995i)22-s + (−0.411 − 0.911i)23-s + (−0.124 + 0.992i)25-s + (0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.863 - 0.504i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.863 - 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5257697684 - 0.1422185686i\)
\(L(\frac12)\) \(\approx\) \(0.5257697684 - 0.1422185686i\)
\(L(1)\) \(\approx\) \(0.5308416171 - 0.06228508674i\)
\(L(1)\) \(\approx\) \(0.5308416171 - 0.06228508674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.998 - 0.0498i)T \)
5 \( 1 + (-0.661 - 0.749i)T \)
11 \( 1 + (-0.998 - 0.0498i)T \)
13 \( 1 + (-0.797 + 0.603i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.411 - 0.911i)T \)
29 \( 1 + (-0.411 + 0.911i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (0.980 - 0.198i)T \)
43 \( 1 + (-0.318 - 0.947i)T \)
47 \( 1 + (0.542 - 0.840i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.0249 + 0.999i)T \)
61 \( 1 + (0.995 - 0.0995i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.826 - 0.563i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
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

−20.70321581538587326212872484899, −20.02659948552049068990734813108, −19.462582674075019024688448506146, −18.63969386327344530185786093388, −17.94534892811525914740867747498, −17.557819211123246584154368537535, −16.26438512692386850383244914723, −15.80032868706014851625700274004, −15.128190071979269719923133525665, −14.40040184489009745215496838154, −13.213868935468620386006249393221, −12.31282034033679170889006216580, −11.24540870389396471660194643043, −11.101200954074525949734726757738, −9.88117990287937779348249409717, −9.53278818921121164594954592656, −8.17825518274107850315888563866, −7.632701134585267499418322448051, −7.14553610105160426071618445661, −6.03505961504388068728283922299, −5.156206531664726010022477465323, −3.8071120652387414871584563172, −2.77537011923901533322091744194, −2.227278290864192323733531194853, −0.59446368947799022051185040244, 0.52012705744886893145943134173, 1.78260012740979947003457868166, 2.66961080373616647108556421912, 3.81528129004713929070509134150, 4.86713943998394752842791535589, 5.74133076897603909392793161086, 7.00101675143743823255040230200, 7.52480889461821091938291521523, 8.41680007906249605249440709882, 9.009037363187760894448449076440, 9.847825750764199872770774070014, 10.762697626696575823591951795486, 11.43263418943120569134769429835, 12.34226176936311443441219057566, 12.812839488819190980180318091136, 14.03588568423456310204114443046, 15.145650371561002868295323854348, 15.66909687891125849515592481111, 16.548210119724639405837338271315, 16.85078033547550348413715245267, 18.05651195584897160331418384885, 18.45991942149162729421965634488, 19.43269498899891665503341511122, 20.09509146679995612935818237930, 20.48411644128911059409655791440

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