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Properties

Label 1-93-93.65-r0-0-0
Degree $1$
Conductor $93$
Sign $-0.101 + 0.994i$
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 + (0.5 + 0.866i)5-s + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (−0.669 + 0.743i)13-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (−0.913 − 0.406i)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 + (0.913 − 0.406i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)10-s + (−0.104 + 0.994i)11-s + (−0.669 + 0.743i)13-s + (0.104 + 0.994i)14-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (−0.913 − 0.406i)22-s + (−0.809 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.101 + 0.994i$
Analytic conductor: \(0.431890\)
Root analytic conductor: \(0.431890\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (0:\ ),\ -0.101 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6026835381 + 0.6673755171i\)
\(L(\frac12)\) \(\approx\) \(0.6026835381 + 0.6673755171i\)
\(L(1)\) \(\approx\) \(0.7913958904 + 0.5105450344i\)
\(L(1)\) \(\approx\) \(0.7913958904 + 0.5105450344i\)

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 \( 1 + (0.913 - 0.406i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.669 - 0.743i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
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

−29.89926704369649986112299365241, −28.92746466669895299789711642589, −28.04487745642488278879977889826, −27.2366330949133736785313439763, −26.07526847476772992486354703707, −24.6786348722925797946851472014, −23.87961592759757982728093011378, −22.075005730744087851075930630528, −21.51154265144797650195994799883, −20.45355542980317803647979965916, −19.57692526158442093646124121440, −18.15708541639123496974919151668, −17.45595348024103911636600918066, −16.29665354061689621899030927101, −14.529921672196092023055534216579, −13.33614353464130358229174644841, −12.362198762515536439519322009203, −11.243980968484035930794302893249, −10.04859920084540309934392334759, −8.75445986292801906366499042714, −8.034048841500905969383165055032, −5.611349131431203900218123177449, −4.533742968769751322873510925259, −2.72021859854272217336729049053, −1.24782455368405231632451555053, 1.958630289365557009377886919917, 4.283550037739425434609848189604, 5.554083083594002894186350143512, 7.03728251148508208373511128616, 7.6930214276678519151427598077, 9.446462058918457618873984338553, 10.26932550293464371164063286218, 11.769650677982327090605424416212, 13.70824257299554116472059327807, 14.350638298614460682818958523654, 15.31134628246626607220357253443, 16.71536124853908781587088174510, 17.80118125753869149024034333577, 18.319101685981570266558749087565, 19.74669017332657080574400960639, 21.1885910236813212424533630706, 22.45441438430707375653110060, 23.2819891068401301035124521254, 24.47999865364606685794106654449, 25.34278321045386726034704339692, 26.48385442328699041423547227572, 27.0520745485588367040276210721, 28.29804789698508960334002307224, 29.519710650886438174125695749714, 30.755297697474379445448778735010

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