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Properties

Label 1-93-93.38-r1-0-0
Degree $1$
Conductor $93$
Sign $0.914 - 0.404i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)10-s + (0.978 − 0.207i)11-s + (−0.104 + 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (0.669 − 0.743i)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 + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)10-s + (0.978 − 0.207i)11-s + (−0.104 + 0.994i)13-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (0.669 − 0.743i)22-s + (−0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(9.99423\)
Root analytic conductor: \(9.99423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 93,\ (1:\ ),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.086377750 - 0.6524478943i\)
\(L(\frac12)\) \(\approx\) \(3.086377750 - 0.6524478943i\)
\(L(1)\) \(\approx\) \(1.938754359 - 0.3786575733i\)
\(L(1)\) \(\approx\) \(1.938754359 - 0.3786575733i\)

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

−30.005139301239575850170066841757, −29.60346127354716234775900874252, −27.845399890021111005014950064583, −26.99750946202572171903098272512, −25.37956044923297989014306836530, −24.954864593426722371512182864084, −23.79588603428822506158881568329, −22.93964918986773568252873009933, −21.66042751725428671772895369881, −20.72468365145942360091229577594, −19.93026532570244324864943048317, −17.77464078964937229647192750705, −17.07222465383271480746310782937, −16.14254775021142207479364742188, −14.64893558888838635384273690443, −13.894917372972209475694126391556, −12.71181612301341913069472176453, −11.754529371135701017694547439708, −10.06751611798631034071384168656, −8.47244440207700780530559706869, −7.44127988257373707920967202739, −5.89084259228207322538528146138, −4.85823724316592774894191904780, −3.59971766565377077306101871970, −1.45487151079067912375177353134, 1.66162904393020026429810638079, 2.87785114442402334947527597925, 4.42885851343741488702100102218, 5.85959123273082726218587568095, 6.857884930585999080988101204023, 8.94373857767245355433344378565, 10.205767495227046878409685532351, 11.415437414889310442295863840060, 12.16607359222846423068688577311, 13.86038097775212187049669667933, 14.43296290415057828986062683907, 15.44968265718251351113944309534, 17.16078772838699031538783644804, 18.57686425474340414034568545428, 19.21782744371515458921432508610, 20.72439976871563765283347141085, 21.752555072404171653761998554165, 22.185411882004211282418484704097, 23.56336146817237116881311453044, 24.57022639490421145234823014233, 25.5822181059467191290761606992, 27.03279708111158694018252984522, 28.147313815468986961003987845747, 29.09098661556858312112219252883, 30.35177620518408626305626486525

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