Properties

Degree 1
Conductor 229
Sign $-0.847 - 0.531i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.847 - 0.531i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.847 - 0.531i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.847 - 0.531i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (34, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ -0.847 - 0.531i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3448470510 - 1.199251110i$
$L(\frac12,\chi)$  $\approx$  $0.3448470510 - 1.199251110i$
$L(\chi,1)$  $\approx$  0.8156518261 - 0.4155666596i
$L(1,\chi)$  $\approx$  0.8156518261 - 0.4155666596i

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

−26.634897779250262243060639990133, −25.85384535140517261698378925713, −25.05552850304801213984906525665, −24.09305215575971567441782086942, −22.49121453952134186541820451792, −21.4117291516035781248159447725, −20.94013076070689712383721126559, −19.99464661812533848649390741560, −18.92920104830344020661865373000, −18.19622572343720096695400673113, −17.28215222191140940978522482045, −16.13071366593393435503162553425, −15.05060340390296836024693862617, −14.469093983026727394991249507359, −13.20789675287864812448043426299, −11.59527127165444032769443930313, −10.75614696889498107618704192978, −9.91854265107097839306811665982, −9.13225346844499813375455806906, −7.69358778219467319972235288848, −7.40047362853785185263439188161, −5.52394653092429328395238046033, −3.99985439644172129862052578359, −2.55401286827627439538135822835, −1.98763650272810497431078053490, 0.45902883128581555000782019746, 1.75455570014505435981161077828, 2.59558440678386727866286531507, 4.73885215354260548519602495914, 5.941796616513511470219113342, 7.31621730092392414883384610064, 8.254208709182664217831574172606, 8.7218780328907387463973175477, 9.88808354479531875753039534036, 11.03877957190088901146590293240, 12.370560125654734812417073552457, 13.21356030048478951447113499956, 14.544488210893518373919960759499, 15.22479695781311381182399486098, 16.47615804354175014992072252135, 17.59975871242822275139304473554, 18.00546327496322175271788205533, 19.2199344564375907332083019779, 20.02959843026343052680811046688, 20.79513341595733323906272248931, 21.543525923484555201946543054189, 23.706298586510878423614394046354, 24.27932936980733252864615492420, 24.70952086898258880233647040084, 25.836460245165852365080291781941

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