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Properties

Label 1-93-93.11-r0-0-0
Degree $1$
Conductor $93$
Sign $0.188 - 0.982i$
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  + (0.809 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ 0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147253220 - 0.9481605123i\)
\(L(\frac12)\) \(\approx\) \(1.147253220 - 0.9481605123i\)
\(L(1)\) \(\approx\) \(1.325382442 - 0.6915264188i\)
\(L(1)\) \(\approx\) \(1.325382442 - 0.6915264188i\)

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

−30.37172222543553783127759379329, −29.8426795450523044334788299963, −28.82970264448318361835692648804, −26.89520796488944874696638166980, −26.24289953384354962089171912392, −25.21189625077995128146534969808, −24.32154876993843096440666250971, −22.90222418066703702254459955013, −22.28024797604471756397563101046, −21.48557037166111198754458505124, −19.97084160258863267450030160468, −18.77417780235760625487736592532, −17.310196902344377239932035262458, −16.494828217841550745294626466394, −15.158025550298454362361833398672, −14.24901358721086692237456907063, −13.27595965114938206578066889868, −12.11122085599234731378076206894, −10.68871150168001399456484214869, −9.32517635263457056730302875296, −7.570406327646563305364846633288, −6.52513483224548923836954688037, −5.591532445609509996461832458490, −3.72832545061314768916565993719, −2.713931073056768087936751565746, 1.55200013428044127451767819918, 3.1464399771849523203599602592, 4.67108647368835285595237293573, 5.74354075273294646928295029238, 7.11804957567405439848525030200, 9.50239788117495179024901506291, 9.72369210737143371461784207589, 11.75365482868840024264460902254, 12.482342993232149681333467552256, 13.500640058523342391297475834552, 14.607842045909079826454964636259, 15.949724624483628857105102883658, 16.984063044614473014573782083403, 18.591880033507568869820674997725, 19.79212148636880679348961634169, 20.47599647848809198938878988070, 21.73509052412672007144030868906, 22.48905020353146807487211536239, 23.623687093755778832426065105513, 24.8270700637013856940062114719, 25.44772524671588738766693510460, 27.280873470272622657933285645888, 28.37684411346419750187722545977, 29.13168663797249887849706648362, 29.88999682126489658539432669668

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