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Properties

Label 1-1323-1323.121-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.989 + 0.143i$
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  + (0.980 − 0.198i)2-s + (0.921 − 0.388i)4-s + (−0.969 + 0.246i)5-s + (0.826 − 0.563i)8-s + (−0.900 + 0.433i)10-s + (0.980 − 0.198i)11-s + (−0.853 + 0.521i)13-s + (0.698 − 0.715i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.797 + 0.603i)20-s + (0.921 − 0.388i)22-s + (−0.124 + 0.992i)23-s + (0.878 − 0.478i)25-s + (−0.733 + 0.680i)26-s + ⋯
L(s)  = 1  + (0.980 − 0.198i)2-s + (0.921 − 0.388i)4-s + (−0.969 + 0.246i)5-s + (0.826 − 0.563i)8-s + (−0.900 + 0.433i)10-s + (0.980 − 0.198i)11-s + (−0.853 + 0.521i)13-s + (0.698 − 0.715i)16-s + (−0.222 + 0.974i)17-s + 19-s + (−0.797 + 0.603i)20-s + (0.921 − 0.388i)22-s + (−0.124 + 0.992i)23-s + (0.878 − 0.478i)25-s + (−0.733 + 0.680i)26-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.628727090 + 0.1891510528i\)
\(L(\frac12)\) \(\approx\) \(2.628727090 + 0.1891510528i\)
\(L(1)\) \(\approx\) \(1.759069757 - 0.05666765427i\)
\(L(1)\) \(\approx\) \(1.759069757 - 0.05666765427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (-0.969 + 0.246i)T \)
11 \( 1 + (0.980 - 0.198i)T \)
13 \( 1 + (-0.853 + 0.521i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.124 + 0.992i)T \)
29 \( 1 + (-0.124 - 0.992i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.698 + 0.715i)T \)
43 \( 1 + (0.270 + 0.962i)T \)
47 \( 1 + (-0.661 - 0.749i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (0.995 - 0.0995i)T \)
61 \( 1 + (0.921 + 0.388i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.853 - 0.521i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−20.67703920386695953327395505253, −20.35874825810782223999149476144, −19.67487883150774086837566103962, −18.865293880282292885938306716057, −17.698648633649078105318817618261, −16.85826323591227643129532711885, −16.16380462615735632452995503444, −15.54815100888072886922402695156, −14.678436875303007574214925488725, −14.22302999003459717816410552960, −13.17028946030003953685219311699, −12.33622630317625239339001276528, −11.86103319881621511874985195459, −11.208515737792794711223285394503, −10.1449500497850187344045935722, −9.064428854496234595166060187078, −8.105936972887609424956536391878, −7.224780712178638514063080034136, −6.817150448095777122452571563872, −5.50750470230846221502844517611, −4.82562404991712438928893599749, −4.03359767779739952861066332404, −3.22703069663005790069877722503, −2.295501308937653970866832006790, −0.86708345920436845304443704703, 1.12488054980660567403523299841, 2.22192389259156402958709684467, 3.3550676827624772494125828361, 3.9150828260485070184704824719, 4.7135701760151147822285675953, 5.712950465819033822296970901177, 6.68472492944895790499827493737, 7.28474775371921697550950784443, 8.16644027762784700743266892743, 9.35713897048046641161631308563, 10.23007694122752195085137343443, 11.26888091860093732407244962386, 11.72323148345959749623618243434, 12.32061805276067636355094684001, 13.25867858047378116675637902137, 14.225943391856153904463824079340, 14.67230123128873741290480777314, 15.48136525087318781684862178427, 16.1675476112883546565899115097, 16.93202701031119056858431538696, 17.91974830606811809045834249819, 19.27540816609816400778834601702, 19.453569921420713559361999799131, 20.037245373235182265455373294274, 21.17896013650439256544513905691

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