↑

Properties

Label 1-173-173.16-r0-0-0
Degree $1$
Conductor $173$
Sign $0.956 - 0.292i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.876847448 - 0.2802295426i\)
\(L(\frac12)\) \(\approx\) \(1.876847448 - 0.2802295426i\)
\(L(1)\) \(\approx\) \(1.677990241 - 0.1683134542i\)
\(L(1)\) \(\approx\) \(1.677990241 - 0.1683134542i\)

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 \( 1 + (0.391 - 0.920i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.76301255290577542101576901792, −26.24419171693598432927674687257, −25.882998275539948481697385492665, −24.27699708052307102100733890816, −23.39444712360303782639931334269, −22.706769537094695446813054786056, −21.82259842584405919727436859719, −20.831423810298510957700660624679, −20.47906990052443160011654175271, −18.66857798221663202048807815251, −17.54540727085980385454386652240, −16.5883500308008739681231440897, −15.57351750945917665102653711764, −14.59797757417207842186763892988, −13.619941290281778659474854025433, −12.69606339165034835845579651778, −11.20340286106969550888578303113, −10.57910958883835362056265730274, −9.89509399726948612739996505071, −7.75877326582451667050311205629, −6.30632026392288331719608776085, −5.71507094864703534488179533481, −4.380480230331056932863850276339, −3.440486282155166816937083883842, −1.74891615499900736021875583065, 1.67928970821861501810628711282, 2.71920138043768147462641438782, 4.84514669999897090429544552666, 5.5405181288302253767714943356, 6.33748041000665972817101994249, 7.66949679994993961632178722426, 8.948900766153493822592721380511, 10.74584860809814908230354201, 11.59817700139675737217418900207, 12.70015462958501450103844601710, 13.30205218186758767401083084809, 14.21273786984288514754390079978, 15.84093280289030566819530504285, 16.30808466178222244252739850674, 17.821464635465934369420598849510, 18.29462131690669753258340722477, 19.871212706992578484345450646385, 21.05140035644040996367143091133, 21.72475740879044737493604639618, 22.62200475721369706219622051999, 23.81538829529494490345792982483, 24.29303058948277389300206780653, 25.24152384581093818418390284628, 25.93943995118423596628501311824, 27.97958827052011549440623956492

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