Properties

Degree 1
Conductor 229
Sign $-0.791 - 0.610i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.791 - 0.610i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.791 - 0.610i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.791 - 0.610i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (106, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ -0.791 - 0.610i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2917682442 - 0.8560689580i$
$L(\frac12,\chi)$  $\approx$  $0.2917682442 - 0.8560689580i$
$L(\chi,1)$  $\approx$  1.044927635 - 0.1488870970i
$L(1,\chi)$  $\approx$  1.044927635 - 0.1488870970i

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

−26.8375099195159255289671501104, −25.35012747619379140933671066861, −24.48671862630393273417333917289, −23.280102936590311629787066399674, −22.58824221682227134946737752950, −22.07428187819259178827246684485, −21.1740146009965982225753916585, −20.33249414224601445165883141272, −18.75549866414143079059629224661, −18.36185434458500335654424806502, −17.09901787472249758897972001662, −15.68676520223940125867773328597, −15.006872844204621222401485318196, −14.352177817486772641074347419897, −12.75140328724751185595533575246, −11.95240825664548047870280696613, −11.25273452133622124038942016775, −10.1556445563719011403440014017, −9.53825760837057655812954172562, −7.32787366197811859951141892576, −6.33171617477285138283756969232, −5.297631380100570863239301452107, −4.48143472217824736344734375329, −3.01076904241228220174933855081, −1.94101700839165145341264566308, 0.23108473359468059866978725996, 1.764328056968689917503096939692, 3.70557222449461314563240648723, 4.96372463817616265062521290767, 5.46142346346900078562503985900, 6.83289185096420511905691969398, 7.64273411229396794373821366696, 8.76751116965009796888215949101, 10.597139646516649617637883434964, 11.47037347101601971787831574807, 12.58052406556522536013047290854, 13.19712599823693890669760988907, 14.03827810605225929744938535346, 15.441987625525722924841493110941, 16.51714861656808419060649507906, 17.06284251623651978257521356190, 17.67895637253762720948351760815, 19.36529009086991504157107981043, 20.29762242758623614144700359356, 21.54311979184514213365458399800, 21.99513346941124244133126235711, 23.42880842334046338687342530068, 23.890190130836451754218433288523, 24.359450749950065733688428553306, 25.430837850679330452420343266199

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