Properties

Degree 1
Conductor 229
Sign $-0.599 + 0.800i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.599 + 0.800i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.599 + 0.800i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.599 + 0.800i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (101, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ -0.599 + 0.800i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.939514814 + 3.873237511i$
$L(\frac12,\chi)$  $\approx$  $1.939514814 + 3.873237511i$
$L(\chi,1)$  $\approx$  1.907592866 + 1.416997816i
$L(1,\chi)$  $\approx$  1.907592866 + 1.416997816i

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

−25.686341531699127915511865114878, −24.79547138072384484282280500104, −23.625045110244775810112644509046, −23.57672740271695459688497592320, −21.840831210583971241410784594435, −21.16544771892386798956961790519, −20.25791236563762198840403099711, −19.526293179102719167761517022529, −18.67674937558264959893803615507, −17.22016728008346339486737110828, −15.928181284543128074209964648634, −15.37462475531694896357472540335, −13.88202413143291845641893650079, −13.115461321534813011380941400242, −12.97790030312767193352522702155, −11.632900274197300700719127068390, −10.23848682934455036795064946331, −9.14057367150015562381112406021, −8.05017729428377018843617200474, −6.6269298353578507801820548436, −5.89291974209184342901057374716, −4.42062746203343607800896544900, −3.28029720555428555532182839154, −2.19187786450513453916217606768, −0.90453089262805417473732101398, 2.38297113395993727112090752092, 3.006675882452092212760668662366, 4.04317521021833551216989679477, 5.41280051622535087426391412810, 6.499746875110782494603320079003, 7.51603840776262877317164852536, 8.77289006939839855630543301174, 10.30121571975536451157334092961, 10.64612317543754416496542345522, 12.34213785975891697076796170759, 13.432933735751240666453092027836, 13.959831753260269481867566229273, 15.0972384589822245277308847698, 15.72675698500032908088943612483, 16.50283703567420534226549799853, 18.120375719904057101054658680349, 19.12091084312528192078987877709, 20.29153571711730078219494673646, 20.98919360089568311685588685850, 21.883270961263901372182446342335, 22.74752289134895605390693303547, 23.29098711282060256428763962212, 25.09397599404762443537758767631, 25.334952440484554902435046753067, 26.24251476241330869587486237382

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