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Properties

Label 1-229-229.123-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.567 + 0.823i$
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  + (−0.837 − 0.546i)2-s + (−0.879 − 0.475i)3-s + (0.401 + 0.915i)4-s + (0.0825 − 0.996i)5-s + (0.475 + 0.879i)6-s + (−0.324 + 0.945i)7-s + (0.164 − 0.986i)8-s + (0.546 + 0.837i)9-s + (−0.614 + 0.789i)10-s + (−0.245 − 0.969i)11-s + (0.0825 − 0.996i)12-s + (0.996 + 0.0825i)13-s + (0.789 − 0.614i)14-s + (−0.546 + 0.837i)15-s + (−0.677 + 0.735i)16-s + (−0.0825 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.837 − 0.546i)2-s + (−0.879 − 0.475i)3-s + (0.401 + 0.915i)4-s + (0.0825 − 0.996i)5-s + (0.475 + 0.879i)6-s + (−0.324 + 0.945i)7-s + (0.164 − 0.986i)8-s + (0.546 + 0.837i)9-s + (−0.614 + 0.789i)10-s + (−0.245 − 0.969i)11-s + (0.0825 − 0.996i)12-s + (0.996 + 0.0825i)13-s + (0.789 − 0.614i)14-s + (−0.546 + 0.837i)15-s + (−0.677 + 0.735i)16-s + (−0.0825 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08450198258 - 0.1609294894i\)
\(L(\frac12)\) \(\approx\) \(-0.08450198258 - 0.1609294894i\)
\(L(1)\) \(\approx\) \(0.4184264306 - 0.2381987279i\)
\(L(1)\) \(\approx\) \(0.4184264306 - 0.2381987279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.837 - 0.546i)T \)
3 \( 1 + (-0.879 - 0.475i)T \)
5 \( 1 + (0.0825 - 0.996i)T \)
7 \( 1 + (-0.324 + 0.945i)T \)
11 \( 1 + (-0.245 - 0.969i)T \)
13 \( 1 + (0.996 + 0.0825i)T \)
17 \( 1 + (-0.0825 + 0.996i)T \)
19 \( 1 + (-0.0825 - 0.996i)T \)
23 \( 1 + (-0.614 - 0.789i)T \)
29 \( 1 + (0.324 - 0.945i)T \)
31 \( 1 + (0.969 - 0.245i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (0.837 + 0.546i)T \)
43 \( 1 + (-0.677 - 0.735i)T \)
47 \( 1 + (-0.837 + 0.546i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (0.735 + 0.677i)T \)
61 \( 1 + (0.546 + 0.837i)T \)
67 \( 1 + (-0.837 + 0.546i)T \)
71 \( 1 + (-0.245 + 0.969i)T \)
73 \( 1 + (-0.164 + 0.986i)T \)
79 \( 1 + (0.324 + 0.945i)T \)
83 \( 1 + (-0.677 + 0.735i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.986 - 0.164i)T \)
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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

−26.74743442546835785098784691107, −26.002929910493579037504081486165, −25.237253707830247349523619099881, −23.691146185360710614615782240392, −23.08145891187594477826218682176, −22.56052576667928128943400575937, −21.01240739969079859781814408264, −20.184547699023062981629368344163, −18.9065150520608417370065414523, −17.97793596804433727270556141690, −17.52027688776583371471220962753, −16.28042402621934310225039382573, −15.73362406368189001179064558017, −14.64902449638905825801967978885, −13.61751176834508245322543392457, −11.914741784458946724754954316555, −10.86992002941220096646356422125, −10.21918504972845696426776244135, −9.551217341875017481822188140420, −7.81238463458265199660795799047, −6.85951456798284978969201502265, −6.19288763369586135541559898681, −4.86889205665746998324106045363, −3.42779576628467340374688170518, −1.469024674497635865482106427196, 0.10123989413445595901505765100, 1.20076450981105046810199198258, 2.47127382997210263974112161450, 4.18549103422870350121423988960, 5.711655491757648995751027469755, 6.49028541087038889793178041396, 8.220200358073363742373200264823, 8.64838852892744168513776646796, 9.97227456489890102129429200796, 11.12307878320158288368966714073, 11.870921788932726906147078135960, 12.81639749147481959979963430563, 13.41033385675147442770849911207, 15.76299216780930298114701750436, 16.17478259361859362491684029594, 17.22715922245593799637162158350, 17.99009833766664115155877772662, 18.97109599771383723212792229574, 19.56199799476350324326249603267, 21.03619491294106324175099771232, 21.547104705568723522560192421097, 22.57327230771311324288742574153, 23.97427088476571847297621014556, 24.56633378280772148641902630529, 25.530730297618064338629122264592

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