Properties

Degree 1
Conductor 229
Sign $0.0240 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $0.0240 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (109, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ 0.0240 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5273843505 - 0.5402065831i$
$L(\frac12,\chi)$  $\approx$  $0.5273843505 - 0.5402065831i$
$L(\chi,1)$  $\approx$  0.5126115015 - 0.2095468060i
$L(1,\chi)$  $\approx$  0.5126115015 - 0.2095468060i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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