↑

Properties

Label 1-229-229.208-r1-0-0
Degree $1$
Conductor $229$
Sign $0.0240 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0240 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0240 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.0240 + 0.999i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.0240 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5273843505 + 0.5402065831i\)
\(L(\frac12)\) \(\approx\) \(0.5273843505 + 0.5402065831i\)
\(L(1)\) \(\approx\) \(0.5126115015 + 0.2095468060i\)
\(L(1)\) \(\approx\) \(0.5126115015 + 0.2095468060i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.996 - 0.0825i)T \)
3 \( 1 + (-0.677 + 0.735i)T \)
5 \( 1 + (-0.789 + 0.614i)T \)
7 \( 1 + (-0.475 + 0.879i)T \)
11 \( 1 + (0.401 - 0.915i)T \)
13 \( 1 + (0.614 + 0.789i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
19 \( 1 + (0.789 + 0.614i)T \)
23 \( 1 + (0.837 + 0.546i)T \)
29 \( 1 + (0.475 - 0.879i)T \)
31 \( 1 + (-0.915 - 0.401i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (0.996 + 0.0825i)T \)
43 \( 1 + (0.945 - 0.324i)T \)
47 \( 1 + (-0.996 + 0.0825i)T \)
53 \( 1 + (-0.677 + 0.735i)T \)
59 \( 1 + (-0.324 + 0.945i)T \)
61 \( 1 + (-0.0825 - 0.996i)T \)
67 \( 1 + (-0.996 + 0.0825i)T \)
71 \( 1 + (0.401 + 0.915i)T \)
73 \( 1 + (0.969 + 0.245i)T \)
79 \( 1 + (0.475 + 0.879i)T \)
83 \( 1 + (0.945 + 0.324i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.245 - 0.969i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.79633158009624144321681098490, −25.0868667511804375380222293903, −24.024362187442056472526220890150, −23.36764958800538080898510061694, −22.55910734295977854618773519001, −20.80930614538967936819773982436, −19.88596809843556557146784827877, −19.452487015047491938059535437335, −18.23961855017372165138750673004, −17.4468370367360979433371423838, −16.57122124499707600274121306484, −15.974024739866514165958212981209, −14.66477073620349437027098242539, −12.969438565130020710144631058669, −12.38889608571529529324967152323, −11.2371438163943062928745480458, −10.465608565845102540097636093179, −9.19754416343034049103758524104, −7.9015928854112729757500702148, −7.3185291963331568385625003981, −6.31334350453227525502654093569, −4.92578927961150513388041802888, −3.27385828677919629268892987372, −1.34479032473586282129685053111, −0.61250342734053242935956496284, 0.841040439711043644426755227, 2.92448397412637834755969893632, 3.78214468551595850728649152369, 5.70198753240378493383217627228, 6.4763653972790625395782748039, 7.74229055303568732307001604107, 9.03830295212298531647843552297, 9.68113669044615585987407844307, 11.0444642334976287403376877397, 11.496801967415648627182145509243, 12.31818758676923281353181623401, 14.35148131370485105975166292671, 15.428598605150203411023961690829, 16.145867292442830210808528493982, 16.67845189085037710817576151308, 18.13491279318515109369794023417, 18.751546420644794688544326384459, 19.52375811888308128225636672993, 20.87343005208478284699434146304, 21.604481824839309522255299674033, 22.5907902710959435523766831879, 23.5220821882151579990812859688, 24.6881464227653864260444790188, 25.77313530651549135862653774275, 26.64417826654273196299637528461

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