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Properties

Label 1-93-93.20-r1-0-0
Degree $1$
Conductor $93$
Sign $-0.729 - 0.684i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)5-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)10-s + (−0.669 − 0.743i)11-s + (0.913 − 0.406i)13-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.669 − 0.743i)20-s + (−0.978 − 0.207i)22-s + (−0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)5-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)10-s + (−0.669 − 0.743i)11-s + (0.913 − 0.406i)13-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.669 − 0.743i)20-s + (−0.978 − 0.207i)22-s + (−0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.729 - 0.684i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ -0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8532789243 - 2.155920076i\)
\(L(\frac12)\) \(\approx\) \(0.8532789243 - 2.155920076i\)
\(L(1)\) \(\approx\) \(1.232997979 - 0.9916298350i\)
\(L(1)\) \(\approx\) \(1.232997979 - 0.9916298350i\)

Euler product

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

−30.70247198439663736043188688025, −29.50289899643670872206547400074, −28.76047439234918153436552588922, −26.86212819815355957082661440640, −25.82128951008750846671967212310, −25.53705776328311146803117789112, −23.97514824733979719701865960985, −22.910869342395958933575564540050, −22.31899521588450562546307799662, −21.185806871902873278008904077332, −20.02971023573311749429598303665, −18.42408670601955960782889642703, −17.515876779305694396430381501033, −16.02384426922126461010597543528, −15.40453019667875389807259551258, −13.85607689880894023375626371511, −13.39232796213848580740382954218, −11.937044962250787749528267328364, −10.575301225639583145132579035940, −9.197631181347154230849709163525, −7.34512223821122480639509203486, −6.605727144843621723574885248915, −5.34823730268362770616945078658, −3.67988065459814246704944252638, −2.51057630812853040749218709993, 0.80940569190139891535422525176, 2.576108328147856771126141710282, 3.968526952211713335981937438991, 5.52452636760870719335903888118, 6.28516530249868565655942927399, 8.49000357803943915255146142671, 9.76778487104001650388442822748, 10.8433243690521840704458413744, 12.36322628784483755286665521904, 13.11941571560459954230238128403, 13.94321357919742457471550119198, 15.669559849626473721656208066580, 16.29228065269991650577085708608, 18.05843692159342656264779889324, 19.23212281189195953889758772074, 20.29942563151336910897922309670, 21.17060043850818447089663591959, 22.15346358945085234566182872654, 23.20738951607536014820611647007, 24.304014844883702720789759218193, 25.15378639790764198223742503839, 26.46725029984904873777733975245, 28.17397058176988054677064076403, 28.71150961327635399028646616724, 29.5140437299770324818220376867

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