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Properties

Label 1-193-193.73-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.169 - 0.985i$
Analytic cond. $20.7407$
Root an. cond. $20.7407$
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  + (0.751 + 0.659i)2-s + (0.382 − 0.923i)3-s + (0.130 + 0.991i)4-s + (−0.986 + 0.162i)5-s + (0.896 − 0.442i)6-s + (−0.258 + 0.965i)7-s + (−0.555 + 0.831i)8-s + (−0.707 − 0.707i)9-s + (−0.849 − 0.528i)10-s + (−0.0980 − 0.995i)11-s + (0.965 + 0.258i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (−0.227 + 0.973i)15-s + (−0.965 + 0.258i)16-s + (−0.352 − 0.935i)17-s + ⋯
L(s)  = 1  + (0.751 + 0.659i)2-s + (0.382 − 0.923i)3-s + (0.130 + 0.991i)4-s + (−0.986 + 0.162i)5-s + (0.896 − 0.442i)6-s + (−0.258 + 0.965i)7-s + (−0.555 + 0.831i)8-s + (−0.707 − 0.707i)9-s + (−0.849 − 0.528i)10-s + (−0.0980 − 0.995i)11-s + (0.965 + 0.258i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (−0.227 + 0.973i)15-s + (−0.965 + 0.258i)16-s + (−0.352 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7787896282 - 0.9244043147i\)
\(L(\frac12)\) \(\approx\) \(0.7787896282 - 0.9244043147i\)
\(L(1)\) \(\approx\) \(1.187392363 + 0.005093662490i\)
\(L(1)\) \(\approx\) \(1.187392363 + 0.005093662490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + (0.751 + 0.659i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (-0.986 + 0.162i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 + (-0.0980 - 0.995i)T \)
13 \( 1 + (0.471 - 0.881i)T \)
17 \( 1 + (-0.352 - 0.935i)T \)
19 \( 1 + (-0.162 - 0.986i)T \)
23 \( 1 + (0.195 - 0.980i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (0.659 - 0.751i)T \)
37 \( 1 + (-0.683 + 0.729i)T \)
41 \( 1 + (-0.910 - 0.412i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.0327 - 0.999i)T \)
53 \( 1 + (-0.999 + 0.0327i)T \)
59 \( 1 + (0.608 + 0.793i)T \)
61 \( 1 + (-0.162 + 0.986i)T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (0.773 - 0.634i)T \)
73 \( 1 + (0.683 + 0.729i)T \)
79 \( 1 + (0.582 + 0.812i)T \)
83 \( 1 + (0.442 - 0.896i)T \)
89 \( 1 + (-0.471 - 0.881i)T \)
97 \( 1 + (0.751 - 0.659i)T \)
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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

−27.14565989195300801815884341141, −26.36687529215271321828986875008, −25.18914340570921378602316524141, −23.70725413289668674143718908223, −23.20852948709863062435318845532, −22.364517140077381479292555352970, −21.12682362907340536877689915479, −20.52812907599875322598609208821, −19.65477925255642125391983707973, −19.05501204796779818937229347324, −17.18825917420861651651680097243, −16.03441037625373137464914150727, −15.30366492245125645592703812331, −14.37934988486514893282133960435, −13.397906182553707856475726450910, −12.223845697055956354159350902547, −11.159964932360915345225413388464, −10.33992431843817519464748065015, −9.377404067470946668673893599544, −7.97319100309248527447576833486, −6.59322232967865131460927304455, −4.92500779960808352594417833348, −4.02273966937359165586500895698, −3.52093761786485892620705313913, −1.72767941971514291748248298949, 0.29764275740108655557454560002, 2.71851094012311805004214365000, 3.32294547762125776197421155607, 5.037252220875827332700551335459, 6.250006456157076597805076063051, 7.11042248846854063380234189325, 8.33601966244272587378148680880, 8.74469961623655256741059167019, 11.1991486873846664070603302873, 11.96376521387761448356482916739, 12.91739923268376470451344844622, 13.71251794028408322870553792182, 14.96354337158886005834975953691, 15.55091736349013037895737471263, 16.58973878203208718128141303451, 18.06612331108713548828322330888, 18.70429398988339629748013568287, 19.8509278362927262902630673870, 20.805251651258589191581075152889, 22.20900631731662918529309253422, 22.81804339054758873608554540047, 23.94728751856416560105207988768, 24.455872160843129390349423566936, 25.36563866554175714462703541358, 26.24394451069828740198752973180

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