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Properties

Label 1-93-93.74-r0-0-0
Degree $1$
Conductor $93$
Sign $0.644 - 0.764i$
Analytic cond. $0.431890$
Root an. cond. $0.431890$
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 + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)10-s + (0.913 + 0.406i)11-s + (0.978 − 0.207i)13-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.913 − 0.406i)20-s + (0.104 − 0.994i)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 + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)10-s + (0.913 + 0.406i)11-s + (0.978 − 0.207i)13-s + (−0.913 + 0.406i)14-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)19-s + (−0.913 − 0.406i)20-s + (0.104 − 0.994i)22-s + (−0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.644 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8177470731 - 0.3801607914i\)
\(L(\frac12)\) \(\approx\) \(0.8177470731 - 0.3801607914i\)
\(L(1)\) \(\approx\) \(0.8849323106 - 0.3191996883i\)
\(L(1)\) \(\approx\) \(0.8849323106 - 0.3191996883i\)

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 \( 1 + (-0.104 - 0.994i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.978 + 0.207i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
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

−30.62304636266016524860946067023, −29.09881731006108173953144707672, −27.98381496821786321735968927050, −27.59326994350795334185365140861, −25.84560359968938992432912008334, −25.29123051757172409392116087487, −24.373388335989287272204135223179, −23.3994779123827629017903519765, −22.04259587737108536816589606513, −21.1057868148072763534427632556, −19.48157266622491133306784243266, −18.54633105827888456499775103699, −17.33694950976199055620342927145, −16.49325337625473610192029919884, −15.48313019883366573843221843868, −14.249196794666479502773095774155, −13.177279977323308502739260458234, −11.894568969909892310811911870527, −10.01368763335637373072316415574, −8.91289269560341479301293854878, −8.24256345662485151465873857281, −6.27467569483574455046850893789, −5.65282994898765282612186143140, −4.0750194780106983438059818027, −1.52517226503800344600175530956, 1.47931506927639906725195393775, 3.136349303167856520592681707384, 4.28161389829273158322913359846, 6.3346705623567714152154786536, 7.6900922197745420583371300563, 9.25286371998944281050767408457, 10.35157981402288467439674478609, 11.07103234401201490557506212230, 12.50288514747104650837582756205, 13.75395938388484376375535827892, 14.50861022434731949222093416663, 16.50600464800180505590917464785, 17.54514975825138872132767284622, 18.44360726739691168401688590641, 19.563229397300104912483642973158, 20.535343551349940145639319207390, 21.58798581047182147741873553821, 22.67177221606443943062097316788, 23.37357559301347061675434257774, 25.40772392167307546696271083082, 26.08540874664364572445482547434, 27.1840764377145755042654380201, 28.05660058778913872897309409113, 29.3943156370061241349051261012, 30.116005290283912241609278559087

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