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Properties

Label 1-229-229.18-r1-0-0
Degree $1$
Conductor $229$
Sign $0.949 + 0.313i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s − 11-s + (0.5 + 0.866i)12-s + i·13-s + (−0.5 + 0.866i)14-s − 15-s + 16-s + 17-s + ⋯
L(s)  = 1  + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 + 0.5i)10-s − 11-s + (0.5 + 0.866i)12-s + i·13-s + (−0.5 + 0.866i)14-s − 15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541326583 + 0.2476383792i\)
\(L(\frac12)\) \(\approx\) \(1.541326583 + 0.2476383792i\)
\(L(1)\) \(\approx\) \(0.9805489077 + 0.1634007418i\)
\(L(1)\) \(\approx\) \(0.9805489077 + 0.1634007418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + iT \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−26.36516633812538922635615464889, −25.60055271032616623084465056864, −23.66713511642478101160451539452, −23.12282781528576340952173319609, −22.16921479267556570378248825951, −21.246579095858008862779949643474, −20.89711259445885706050358242730, −19.73821295934383851701979847797, −18.40651278922604669098381786969, −17.72919544936246766323264849357, −17.05060162687259243313362994457, −15.42418218246362629848114708300, −14.59085131198145714990369896941, −13.636467373871094938872493554630, −12.44520189235823133790030009252, −11.12878704270387810154823926310, −10.644846920015393479361682849208, −10.038476199270992950125367855346, −8.74320257554863002919875243396, −7.41027325224182254937769511343, −5.58901492350218809694000031701, −4.944464774253331704401317197834, −3.54526198333727623619369889884, −2.60209424755330302224822426273, −0.85287982540316697166943636234, 0.83171305193853119078454076612, 2.124911092392645373112065340528, 4.5077378595081239574756959839, 5.413051263677394386437052677236, 6.0765805845729720282918532673, 7.49278031947630061497022327188, 8.233534263001997974911328568243, 9.181277144688246854754797597571, 10.632010382950705944828264989432, 12.14447620411147898881062033493, 12.774711021729745864883258009766, 13.84327645074249332385537022687, 14.60221965387975908204758678998, 16.06502087157164413497139293710, 16.79044726774390308622003722811, 17.58489489469661633432325502770, 18.44834453106260878879602719171, 19.11648750889115145464054656728, 20.93993027090744267233923225648, 21.50274249538198415246310297005, 22.92952622643605762921899050633, 23.6959585034189991236935783637, 24.318273301482358291251442420614, 25.07066837521490072079994600762, 25.78192346192675246041778451648

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