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Properties

Label 1-173-173.10-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.990 + 0.139i$
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  + (0.109 + 0.994i)2-s + (−0.181 + 0.983i)3-s + (−0.976 + 0.217i)4-s + (0.905 + 0.424i)5-s + (−0.997 − 0.0729i)6-s + (0.833 + 0.551i)7-s + (−0.322 − 0.946i)8-s + (−0.934 − 0.357i)9-s + (−0.322 + 0.946i)10-s + (−0.581 + 0.813i)11-s + (−0.0365 − 0.999i)12-s + (0.109 + 0.994i)13-s + (−0.457 + 0.889i)14-s + (−0.581 + 0.813i)15-s + (0.905 − 0.424i)16-s + (0.989 − 0.145i)17-s + ⋯
L(s)  = 1  + (0.109 + 0.994i)2-s + (−0.181 + 0.983i)3-s + (−0.976 + 0.217i)4-s + (0.905 + 0.424i)5-s + (−0.997 − 0.0729i)6-s + (0.833 + 0.551i)7-s + (−0.322 − 0.946i)8-s + (−0.934 − 0.357i)9-s + (−0.322 + 0.946i)10-s + (−0.581 + 0.813i)11-s + (−0.0365 − 0.999i)12-s + (0.109 + 0.994i)13-s + (−0.457 + 0.889i)14-s + (−0.581 + 0.813i)15-s + (0.905 − 0.424i)16-s + (0.989 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08134271924 + 1.157161333i\)
\(L(\frac12)\) \(\approx\) \(0.08134271924 + 1.157161333i\)
\(L(1)\) \(\approx\) \(0.6062372494 + 0.9148460501i\)
\(L(1)\) \(\approx\) \(0.6062372494 + 0.9148460501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.109 + 0.994i)T \)
3 \( 1 + (-0.181 + 0.983i)T \)
5 \( 1 + (0.905 + 0.424i)T \)
7 \( 1 + (0.833 + 0.551i)T \)
11 \( 1 + (-0.581 + 0.813i)T \)
13 \( 1 + (0.109 + 0.994i)T \)
17 \( 1 + (0.989 - 0.145i)T \)
19 \( 1 + (-0.457 - 0.889i)T \)
23 \( 1 + (-0.581 - 0.813i)T \)
29 \( 1 + (-0.997 + 0.0729i)T \)
31 \( 1 + (-0.181 - 0.983i)T \)
37 \( 1 + (-0.457 - 0.889i)T \)
41 \( 1 + (0.833 + 0.551i)T \)
43 \( 1 + (-0.976 - 0.217i)T \)
47 \( 1 + (0.744 - 0.667i)T \)
53 \( 1 + (0.957 + 0.288i)T \)
59 \( 1 + (0.391 + 0.920i)T \)
61 \( 1 + (0.989 + 0.145i)T \)
67 \( 1 + (-0.181 + 0.983i)T \)
71 \( 1 + (-0.997 + 0.0729i)T \)
73 \( 1 + (-0.791 - 0.611i)T \)
79 \( 1 + (0.744 + 0.667i)T \)
83 \( 1 + (0.639 - 0.768i)T \)
89 \( 1 + (0.520 - 0.853i)T \)
97 \( 1 + (0.639 + 0.768i)T \)
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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

−27.42458456285784835792030774153, −26.04778215043670694977643199707, −24.98117234907070172434190373233, −23.920215473262392081764258743037, −23.295668502093105917099124099112, −22.06731963873453897610478038976, −20.9916251297986978365892332187, −20.4149835595679134819413310242, −19.20230142087816233193267416930, −18.2470321231737879316852817033, −17.57399858110388136374068121150, −16.6987225936278587044025024458, −14.5333379001638908216110468101, −13.74948970817455584225824249837, −13.0234942618618620721373148929, −12.09924003443830133605299033027, −10.90747629341932825628870592756, −10.11499868764323666379078040863, −8.53431354926066744684775674255, −7.81030305797533990637799967091, −5.81679514839609494044505278660, −5.23865543696060230812697874461, −3.38999435708281690962328919806, −1.9574239401086123146839465647, −1.022872460052649152084563703793, 2.3605317328069462345835963500, 4.14636055500331753796838550232, 5.165279886295615645837185257197, 5.95067277697210728581061603782, 7.28810761036677276111583208851, 8.71443391670331415505373287110, 9.5376767422041991179368767990, 10.537748998501424679732167438030, 11.911099805436132444182218766687, 13.37254472736845146208582431534, 14.63534853009488781283421298676, 14.8460975060784987689430964511, 16.14286314167010268599329220147, 17.06467198617981781225088100761, 17.90413732004240172335433210257, 18.70830957146580868456135155136, 20.69448852570874004699814828192, 21.42536014798637225610497583189, 22.116944479985488929275601488868, 23.1257358820742100720689631064, 24.128958938082058737314297279177, 25.28884451967867131724742565913, 26.07168901344320714541436843834, 26.6236510511506964194942593442, 28.12319615193520322186044731986

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