Properties

Degree 1
Conductor $ 3 \cdot 31 $
Sign $0.308 + 0.951i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.5 + 0.866i)5-s + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)10-s + (0.104 − 0.994i)11-s + (0.669 − 0.743i)13-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (0.104 + 0.994i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.913 + 0.406i)22-s + (0.809 − 0.587i)23-s + ⋯
L(s,χ)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.5 + 0.866i)5-s + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)10-s + (0.104 − 0.994i)11-s + (0.669 − 0.743i)13-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (0.104 + 0.994i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.913 + 0.406i)22-s + (0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.308 + 0.951i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.308 + 0.951i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(93\)    =    \(3 \cdot 31\)
\( \varepsilon \)  =  $0.308 + 0.951i$
motivic weight  =  \(0\)
character  :  $\chi_{93} (59, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 93,\ (1:\ ),\ 0.308 + 0.951i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.355421051 + 0.9855105445i$
$L(\frac12,\chi)$  $\approx$  $1.355421051 + 0.9855105445i$
$L(\chi,1)$  $\approx$  1.004798479 + 0.5024054036i
$L(1,\chi)$  $\approx$  1.004798479 + 0.5024054036i

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

−29.74951774798204875030212336989, −28.52498165456354679567420205984, −28.12546385284803961805033890710, −27.022106413675827369979539757753, −25.72219828991588912555043297519, −24.69756203686506415732786101647, −23.41014877619823327507216565218, −22.13798256546795180331913305354, −20.88156804797443327874163566328, −20.67591875290164726552228309725, −19.23436413697557870982424486514, −17.93521327780911993553139591895, −17.38862770362879957130889312663, −15.9567580810550543578612162473, −14.238931152781543200707290565661, −13.22618030672355359572375503351, −12.04141344919678500499340622462, −11.19100269317147973352990666374, −9.549574856620137996414649349069, −8.93972913917301175272344173359, −7.519873247443365282918711524094, −5.29060490768283521764685539615, −4.33110955614134678933807844468, −2.3192269202154278593883668817, −1.14925444556479764746698725051, 1.25789081481102820777942792202, 3.56255099453710882211583454144, 5.35304236970681056014002432659, 6.356325876535264512698561509317, 7.68365236451526006015633337915, 8.680885843414025616554586249159, 10.28775026116339889370287373953, 11.03750233070061724999840549382, 13.19320783208249486914779025624, 14.235084857935837655042231987047, 14.91173106877975521888984548253, 16.3119943030689120655881383840, 17.41759685095387721079680127083, 18.24208002160720466452179888728, 19.16728330832350536588668967675, 20.81079351670426717484146769497, 22.037297162111749296651874816918, 23.043704507027235106425220457, 24.1118369700445835558962287451, 25.04856838766954878778620088583, 26.12615260427789803882221425185, 26.93036414772877259832251569686, 27.78588143199944130531659696356, 29.218780727864015461628574791647, 30.33243242069269190406698772050

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