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Properties

Label 1-93-93.2-r1-0-0
Degree $1$
Conductor $93$
Sign $0.800 - 0.599i$
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 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9474046375 - 0.3156101340i\)
\(L(\frac12)\) \(\approx\) \(0.9474046375 - 0.3156101340i\)
\(L(1)\) \(\approx\) \(0.7248521035 - 0.2474525719i\)
\(L(1)\) \(\approx\) \(0.7248521035 - 0.2474525719i\)

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

−30.36527925567848618368152648066, −28.80272140589625461476314037710, −27.82181395384397324280158707555, −26.865188308445952137765799132845, −26.045983538795360179668268173140, −24.996019444854689645826960117185, −23.73489094430260820489966948075, −23.10518517526850657673420653211, −22.15876155051062778758424982902, −20.20148274064976527033300775639, −19.34231875757974158077423975975, −18.39520740196344584960439111303, −16.839206970703987617872238657727, −16.28721936874522215412057925346, −15.1157852441330143438500561331, −14.07821539329489495883117200624, −12.76176248744564588878880265121, −11.31254838267728442464876800376, −9.7800562279187729298514694241, −8.77843919160778307647819549197, −7.253255402093452512131120675895, −6.7037999160727186873616526231, −4.79687455819982093886254091822, −3.64764265096860208015450873549, −0.7751747836649751171430679065, 0.92178707328026510841821818920, 3.0616868796664639147116316828, 3.87974187667603770256243470558, 5.7755353898485421819419275133, 7.70076684325265944131690279373, 8.73981707617845899438233913452, 9.95468260748994432004088938136, 11.25895940602520426265620099752, 12.18603678665267880538413308135, 13.094493031463240159191827098848, 14.679649479541786617452685280411, 16.02002363523219375983954561115, 17.1006855464272107060116774708, 18.663586993986611289748213293588, 19.22412956651429967210719413921, 20.159304276015440055487245209694, 21.383689265269696322446795352041, 22.5584146116521778554855641590, 23.153934405266608635760818630863, 24.84103612461276965724900167306, 25.980972466016037050392001984908, 27.24217741888626379294205044858, 27.74815390616604280761729671605, 28.88799906951735747320614862346, 29.921331312385274372462174720961

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