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Properties

Label 1-229-229.218-r0-0-0
Degree $1$
Conductor $229$
Sign $-0.961 + 0.274i$
Analytic cond. $1.06347$
Root an. cond. $1.06347$
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  + (−0.879 + 0.475i)2-s + (0.245 − 0.969i)3-s + (0.546 − 0.837i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (−0.986 − 0.164i)7-s + (−0.0825 + 0.996i)8-s + (−0.879 − 0.475i)9-s + (0.945 + 0.324i)10-s + (0.789 + 0.614i)11-s + (−0.677 − 0.735i)12-s + (−0.677 − 0.735i)13-s + (0.945 − 0.324i)14-s + (−0.879 + 0.475i)15-s + (−0.401 − 0.915i)16-s + (−0.677 − 0.735i)17-s + ⋯
L(s)  = 1  + (−0.879 + 0.475i)2-s + (0.245 − 0.969i)3-s + (0.546 − 0.837i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (−0.986 − 0.164i)7-s + (−0.0825 + 0.996i)8-s + (−0.879 − 0.475i)9-s + (0.945 + 0.324i)10-s + (0.789 + 0.614i)11-s + (−0.677 − 0.735i)12-s + (−0.677 − 0.735i)13-s + (0.945 − 0.324i)14-s + (−0.879 + 0.475i)15-s + (−0.401 − 0.915i)16-s + (−0.677 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ -0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02776265765 - 0.1987631870i\)
\(L(\frac12)\) \(\approx\) \(0.02776265765 - 0.1987631870i\)
\(L(1)\) \(\approx\) \(0.4423747636 - 0.1778003356i\)
\(L(1)\) \(\approx\) \(0.4423747636 - 0.1778003356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.879 + 0.475i)T \)
3 \( 1 + (0.245 - 0.969i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.986 - 0.164i)T \)
11 \( 1 + (0.789 + 0.614i)T \)
13 \( 1 + (-0.677 - 0.735i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
19 \( 1 + (-0.677 + 0.735i)T \)
23 \( 1 + (0.945 - 0.324i)T \)
29 \( 1 + (-0.986 - 0.164i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.879 + 0.475i)T \)
43 \( 1 + (-0.401 + 0.915i)T \)
47 \( 1 + (-0.879 - 0.475i)T \)
53 \( 1 + (0.245 - 0.969i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (-0.879 - 0.475i)T \)
71 \( 1 + (0.789 - 0.614i)T \)
73 \( 1 + (-0.0825 + 0.996i)T \)
79 \( 1 + (-0.986 + 0.164i)T \)
83 \( 1 + (-0.401 - 0.915i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.0825 - 0.996i)T \)
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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

−26.79416937483235804902299226878, −26.274839480466781743496163050398, −25.56092455575733949626885786618, −24.28219966066702912527678304107, −22.72746141037070878304779251822, −21.99479219411189820538800167915, −21.412069230887771849287345343144, −19.98141003647769329635609899491, −19.38060198336685974371456814300, −18.90434967312598510692783330778, −17.24096836524880960230039103758, −16.617330791951515495237247331992, −15.56001374587608477420554105451, −14.919675216723881387584595275285, −13.44686932366064378786426050842, −12.02632964735204809795888927524, −11.157347859333232920467436089868, −10.40192930587241080600465044332, −9.25907709483615433775218797352, −8.715292722404612273306726563609, −7.238746574219005942697222348162, −6.29845412600000367584412122776, −4.20546032214185271595054529635, −3.406707801288707526171122807496, −2.399767577001725023851222329251, 0.18213035159659015731390839846, 1.587519875306744757039951274192, 3.10872104873736229240570543679, 4.921657185302000883039079374072, 6.41505027452500848022655917371, 7.10130166095253886235586122310, 8.10344753409360599835162156000, 9.0075751421027385988100631195, 9.92601828321495428396209605425, 11.4680199180115054228222693822, 12.38492660691425179322392103152, 13.26163631063021665537870294124, 14.680767462655830521592924438942, 15.47075458526552498156816650876, 16.7373431350866053845820410737, 17.213359923793603366730339822307, 18.4425769891874197898498972231, 19.3533811173694356860333712558, 19.89130099423366997687339539775, 20.56044896796384710285964281865, 22.7677040347304218017367292456, 23.145074618918417352146309940962, 24.47312084993084762658495646023, 24.86955019778353064044840429114, 25.672912261881617978718640136277

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