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Properties

Label 1-93-93.41-r1-0-0
Degree $1$
Conductor $93$
Sign $0.308 - 0.951i$
Analytic cond. $9.99423$
Root an. cond. $9.99423$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355421051 - 0.9855105445i\)
\(L(\frac12)\) \(\approx\) \(1.355421051 - 0.9855105445i\)
\(L(1)\) \(\approx\) \(1.004798479 - 0.5024054036i\)
\(L(1)\) \(\approx\) \(1.004798479 - 0.5024054036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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