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Properties

Label 1-93-93.29-r0-0-0
Degree $1$
Conductor $93$
Sign $0.0229 + 0.999i$
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 − 5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (0.809 + 0.587i)22-s + (−0.809 − 0.587i)23-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s − 5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + (0.809 + 0.587i)22-s + (−0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0229 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0229 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.0229 + 0.999i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.0229 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1794262832 + 0.1753608696i\)
\(L(\frac12)\) \(\approx\) \(0.1794262832 + 0.1753608696i\)
\(L(1)\) \(\approx\) \(0.4885749216 - 0.06182331283i\)
\(L(1)\) \(\approx\) \(0.4885749216 - 0.06182331283i\)

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 \( 1 - T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
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

−30.12151048748515789405761691248, −28.72356000060964689270575292085, −27.76371011361865124218801283810, −26.54739773034467209918759028384, −26.23011411724047458615736147312, −24.70992663860124363204930991908, −23.77052600165054090354563146919, −23.00497798621241589395950229371, −21.98680385686452974199320481944, −19.99067440858197821789097493786, −19.393843466654275573141680134708, −18.13874377601095441223459600408, −16.97920294305469572982339446631, −15.777223895821587555118430367807, −15.38054840281575527271615897288, −13.72227757854663545951599453828, −12.80833001595303596384912697889, −10.99684847004880930191366293407, −9.87641558856367981902158755950, −8.37296245490931825170303592054, −7.51716743119104837085681185422, −6.30042438049904673296144113382, −4.79029752198621488098551207281, −3.37690498987978291300532040551, −0.28095298460707008543435670870, 2.26345729478345148971486820037, 3.60877027362859901847280488403, 4.901401603329683310384305007128, 7.022669876476904650215214288769, 8.352244380417157321194311379127, 9.47797947046959041438806226066, 10.67454892880025466274307744147, 12.01522515189595223019504250628, 12.53532469333846294491311804490, 14.02548421636915032376539187185, 15.606934429500305843657226928938, 16.53059883160064166223888383327, 18.1720748884515142922576137906, 18.90546032333512814493855989423, 19.87487010176595454725962311426, 20.800414135952987515357002835265, 22.16560409546848591418605914772, 22.86291001815744731828679219012, 24.076745750486762731373560856371, 25.71950775916234521520860846922, 26.582369513757688365805669414999, 27.56710377462643170093049332657, 28.655075613106022487452504605484, 29.18824973207093617719245433760, 30.76181912541908545093977842475

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