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Properties

Label 1-173-173.54-r0-0-0
Degree $1$
Conductor $173$
Sign $0.214 - 0.976i$
Analytic cond. $0.803408$
Root an. cond. $0.803408$
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  + (−0.989 + 0.145i)2-s + (0.694 − 0.719i)3-s + (0.957 − 0.288i)4-s + (−0.833 − 0.551i)5-s + (−0.581 + 0.813i)6-s + (−0.252 + 0.967i)7-s + (−0.905 + 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (0.976 − 0.217i)11-s + (0.457 − 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (0.322 − 0.946i)17-s + ⋯
L(s)  = 1  + (−0.989 + 0.145i)2-s + (0.694 − 0.719i)3-s + (0.957 − 0.288i)4-s + (−0.833 − 0.551i)5-s + (−0.581 + 0.813i)6-s + (−0.252 + 0.967i)7-s + (−0.905 + 0.424i)8-s + (−0.0365 − 0.999i)9-s + (0.905 + 0.424i)10-s + (0.976 − 0.217i)11-s + (0.457 − 0.889i)12-s + (0.989 − 0.145i)13-s + (0.109 − 0.994i)14-s + (−0.976 + 0.217i)15-s + (0.833 − 0.551i)16-s + (0.322 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.214 - 0.976i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.214 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6184085008 - 0.4971548785i\)
\(L(\frac12)\) \(\approx\) \(0.6184085008 - 0.4971548785i\)
\(L(1)\) \(\approx\) \(0.7410687735 - 0.2607641046i\)
\(L(1)\) \(\approx\) \(0.7410687735 - 0.2607641046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.989 + 0.145i)T \)
3 \( 1 + (0.694 - 0.719i)T \)
5 \( 1 + (-0.833 - 0.551i)T \)
7 \( 1 + (-0.252 + 0.967i)T \)
11 \( 1 + (0.976 - 0.217i)T \)
13 \( 1 + (0.989 - 0.145i)T \)
17 \( 1 + (0.322 - 0.946i)T \)
19 \( 1 + (-0.109 - 0.994i)T \)
23 \( 1 + (-0.976 - 0.217i)T \)
29 \( 1 + (-0.581 - 0.813i)T \)
31 \( 1 + (-0.694 - 0.719i)T \)
37 \( 1 + (0.109 + 0.994i)T \)
41 \( 1 + (0.252 - 0.967i)T \)
43 \( 1 + (0.957 + 0.288i)T \)
47 \( 1 + (-0.997 - 0.0729i)T \)
53 \( 1 + (0.791 - 0.611i)T \)
59 \( 1 + (0.872 + 0.489i)T \)
61 \( 1 + (0.322 + 0.946i)T \)
67 \( 1 + (-0.694 + 0.719i)T \)
71 \( 1 + (0.581 + 0.813i)T \)
73 \( 1 + (0.639 + 0.768i)T \)
79 \( 1 + (0.997 - 0.0729i)T \)
83 \( 1 + (0.391 - 0.920i)T \)
89 \( 1 + (0.744 + 0.667i)T \)
97 \( 1 + (-0.391 - 0.920i)T \)
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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

−27.56276883584828244253210775472, −26.74899317327818161215423834876, −26.01710819732143840138916194968, −25.35963550693322656199347445088, −23.95017014296941126846975476713, −22.833829430239479016780697407066, −21.657655490954000312485516905565, −20.551309775055080334536398537383, −19.780299259220716127665078660216, −19.24907419035398520620298162449, −18.0500270693028219799935207571, −16.607192527023211148839324259495, −16.17405933508675260130726053665, −14.94455065742587242600165201661, −14.168435173549221604036149490986, −12.487208622581610933522411008064, −11.09763288068638687238022230981, −10.51017049169835028672245094838, −9.45160927545054649018823597169, −8.30818092409795279945331226011, −7.507922225867234381976028845183, −6.31242895870243611509879614554, −3.82482801190383537978987060871, −3.589650608077073913132161133947, −1.66196961330314016477414825564, 0.885158022172337517242732315072, 2.35554801054246730978071218684, 3.67949387565887868106700940667, 5.81502299586467511746437435373, 6.91431389670751079248601042464, 8.04104278963309820292263014384, 8.823265174517199560772652363210, 9.47491161818611898878272608977, 11.46770743418195172502839736924, 11.94863276359665904263636668385, 13.211394938870478842676341447964, 14.67412921232382197530187750675, 15.57716466181677795854174936886, 16.391792651428133266906122553438, 17.74908151096491099065088020067, 18.6749988458240108981209134698, 19.32298203217730205599674953662, 20.16946519809400533092970543275, 20.96862270607582432119416111606, 22.61889556526429132013838548955, 24.02428014551065880307981390109, 24.49153353391565041923135503630, 25.44740751153277826869682638791, 26.133774099547005973764027159537, 27.45534740167061387394986987553

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