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Properties

Label 1-93-93.50-r1-0-0
Degree $1$
Conductor $93$
Sign $-0.630 + 0.776i$
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.630 + 0.776i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ -0.630 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005558291565 + 0.01166683459i\)
\(L(\frac12)\) \(\approx\) \(0.005558291565 + 0.01166683459i\)
\(L(1)\) \(\approx\) \(0.5938845611 - 0.2121069259i\)
\(L(1)\) \(\approx\) \(0.5938845611 - 0.2121069259i\)

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

−29.25993810952835631996619308649, −28.46827549244191093544984471667, −27.59935949630121829764511786685, −26.37415384081763357144104453989, −25.243305729772951910826658329047, −24.6932772580986318585105026053, −23.669004173995898042268936919112, −22.386898561995935249944944877271, −21.33789592960998614714350747103, −19.95663513016891643102958119272, −18.6498183083804657683618095564, −17.71363292664262093762728745616, −16.72373244172530224135034435327, −15.626357829270661049721817051474, −14.74342184162301442208712182806, −13.229955317939551006514954234355, −12.4454204408456472321344751335, −10.38877811405822657937261665975, −9.19559406609643197325558267103, −8.41618822904344536720458020344, −6.92599735263876091626181513959, −5.49444624977054215329595149448, −4.76261191272912682337022737277, −2.1724095692534424987816255601, −0.00605021043322642951353618967, 2.05063365429316124340147553480, 3.300662352438566625524433734614, 4.79783862561646045659753768281, 6.74397441786251770123817683439, 8.01850012342780732351921427280, 9.62643860996042730630912969050, 10.52225988912205563685219032415, 11.30699748872698913277930483803, 13.02927941448142059781309303423, 13.69450131742151982451858061129, 15.04522057112433590445763388032, 16.92193268013687968626973895664, 17.6190505092553781877691477249, 18.86523204653341764717385632795, 19.644682751404061528789749274902, 20.93920669545366994772780750640, 21.77935204144270979609562725038, 22.787368141711032997399870064581, 23.85968528419801483722984387055, 25.650239385759572070551234997501, 26.543457995506152105912192794036, 27.06971228453859791772418540270, 28.67711397493099388631192594261, 29.424263588935369619975964941721, 30.12323548115151142819560768641

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