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Properties

Label 1-173-173.160-r0-0-0
Degree $1$
Conductor $173$
Sign $0.698 + 0.715i$
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  + (−0.0365 + 0.999i)2-s + (0.833 − 0.551i)3-s + (−0.997 − 0.0729i)4-s + (0.989 − 0.145i)5-s + (0.520 + 0.853i)6-s + (−0.322 + 0.946i)7-s + (0.109 − 0.994i)8-s + (0.391 − 0.920i)9-s + (0.109 + 0.994i)10-s + (0.744 − 0.667i)11-s + (−0.872 + 0.489i)12-s + (−0.0365 + 0.999i)13-s + (−0.934 − 0.357i)14-s + (0.744 − 0.667i)15-s + (0.989 + 0.145i)16-s + (−0.457 − 0.889i)17-s + ⋯
L(s)  = 1  + (−0.0365 + 0.999i)2-s + (0.833 − 0.551i)3-s + (−0.997 − 0.0729i)4-s + (0.989 − 0.145i)5-s + (0.520 + 0.853i)6-s + (−0.322 + 0.946i)7-s + (0.109 − 0.994i)8-s + (0.391 − 0.920i)9-s + (0.109 + 0.994i)10-s + (0.744 − 0.667i)11-s + (−0.872 + 0.489i)12-s + (−0.0365 + 0.999i)13-s + (−0.934 − 0.357i)14-s + (0.744 − 0.667i)15-s + (0.989 + 0.145i)16-s + (−0.457 − 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389595934 + 0.5857742994i\)
\(L(\frac12)\) \(\approx\) \(1.389595934 + 0.5857742994i\)
\(L(1)\) \(\approx\) \(1.290896628 + 0.4263618188i\)
\(L(1)\) \(\approx\) \(1.290896628 + 0.4263618188i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.0365 + 0.999i)T \)
3 \( 1 + (0.833 - 0.551i)T \)
5 \( 1 + (0.989 - 0.145i)T \)
7 \( 1 + (-0.322 + 0.946i)T \)
11 \( 1 + (0.744 - 0.667i)T \)
13 \( 1 + (-0.0365 + 0.999i)T \)
17 \( 1 + (-0.457 - 0.889i)T \)
19 \( 1 + (-0.934 + 0.357i)T \)
23 \( 1 + (0.744 + 0.667i)T \)
29 \( 1 + (0.520 - 0.853i)T \)
31 \( 1 + (0.833 + 0.551i)T \)
37 \( 1 + (-0.934 + 0.357i)T \)
41 \( 1 + (-0.322 + 0.946i)T \)
43 \( 1 + (-0.997 + 0.0729i)T \)
47 \( 1 + (-0.694 + 0.719i)T \)
53 \( 1 + (-0.581 - 0.813i)T \)
59 \( 1 + (-0.791 - 0.611i)T \)
61 \( 1 + (-0.457 + 0.889i)T \)
67 \( 1 + (0.833 - 0.551i)T \)
71 \( 1 + (0.520 - 0.853i)T \)
73 \( 1 + (-0.976 + 0.217i)T \)
79 \( 1 + (-0.694 - 0.719i)T \)
83 \( 1 + (0.957 + 0.288i)T \)
89 \( 1 + (-0.181 - 0.983i)T \)
97 \( 1 + (0.957 - 0.288i)T \)
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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

−27.44810298368609154304374368125, −26.39053686024631393479921415623, −25.80518518646440381072613882074, −24.73769030726253915211590096158, −23.10120353331176312951843327408, −22.2356234581499515815310265070, −21.43429503302778216209413162536, −20.44619735044924063190080222887, −19.90278142701834994653979468603, −18.92724075239749114108328671815, −17.49781396907047550074041717582, −16.9764247031496231341900249060, −15.13551625071213500792123802536, −14.27168936918323999613912141038, −13.33530793931696557183594572477, −12.65495019789898198426937777817, −10.65757015337756773177251236015, −10.34791245898068071175108101069, −9.30630981034596034643852042516, −8.36492638209456376083970425394, −6.74288480995056073616659515480, −4.95552289213290024016196632924, −3.91585705894219280150947439176, −2.77810452801098250538258621156, −1.58126828938857933235266365609, 1.61799410220092073132637043633, 3.11262933383163487938854331672, 4.770622375794647905424476192935, 6.27333290200494834555023195851, 6.67595668106866445754570646573, 8.3588815090206023131112666072, 9.063358910165031913297502971409, 9.72788474135076287605245531745, 11.890181113235005092519147853654, 13.107973391339886324514635603884, 13.83416462314569915734882946472, 14.61022285900357963274749325628, 15.6821071455125587612487839044, 16.833424115845691087279935099433, 17.8106298441643659959930984528, 18.8002308429464731196221288374, 19.378243713973782680281284749806, 21.14819284172337562621182097074, 21.735783830621288624373245837307, 22.97967420032907528294591980039, 24.26561559185247256102684780742, 24.90040299981994444308635240424, 25.400083931474823537239243525685, 26.32808111872176087962521668055, 27.267618696518972463640820897852

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