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Properties

Label 1-173-173.5-r1-0-0
Degree $1$
Conductor $173$
Sign $-0.375 - 0.926i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
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  + (0.145 + 0.989i)2-s + (0.719 + 0.694i)3-s + (−0.957 + 0.288i)4-s + (0.551 − 0.833i)5-s + (−0.581 + 0.813i)6-s + (−0.967 − 0.252i)7-s + (−0.424 − 0.905i)8-s + (0.0365 + 0.999i)9-s + (0.905 + 0.424i)10-s + (0.217 + 0.976i)11-s + (−0.889 − 0.457i)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 + (−0.946 − 0.322i)17-s + ⋯
L(s)  = 1  + (0.145 + 0.989i)2-s + (0.719 + 0.694i)3-s + (−0.957 + 0.288i)4-s + (0.551 − 0.833i)5-s + (−0.581 + 0.813i)6-s + (−0.967 − 0.252i)7-s + (−0.424 − 0.905i)8-s + (0.0365 + 0.999i)9-s + (0.905 + 0.424i)10-s + (0.217 + 0.976i)11-s + (−0.889 − 0.457i)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 + (−0.946 − 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.375 - 0.926i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ -0.375 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2390796228 + 0.3546719658i\)
\(L(\frac12)\) \(\approx\) \(-0.2390796228 + 0.3546719658i\)
\(L(1)\) \(\approx\) \(0.7164837256 + 0.5889299542i\)
\(L(1)\) \(\approx\) \(0.7164837256 + 0.5889299542i\)

Euler product

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

−26.370710153306917526173679332112, −25.9592742807123321494021137935, −24.625508212832758037505554660696, −23.66444208471388720010142719302, −22.24695302605249006214572138217, −21.96191218790375605427100452594, −20.66980622054308332371075997615, −19.36167558606358954534333728016, −19.24069290716392183734521779498, −18.14850318609746879102890372677, −17.1783611396067978212024969607, −15.26648647925323932517785296152, −14.32581716322255190861221614961, −13.450511939939319559875326374146, −12.736084316031437415550449559272, −11.565872517800283263675424929710, −10.30137522035947324868159046252, −9.40594539396838759719869110181, −8.40567329241426223490012364925, −6.78820930412644789233966761493, −5.84916262721328418138110456444, −3.82408289580731300448006175287, −2.81213051420108572499374091860, −2.01530046831295775918184609695, −0.12200828245296004936295235926, 2.347855216831684376275995198041, 4.11630751158695460182202552637, 4.74540903686573983508051654767, 6.17003432920261229591671649561, 7.38257433707958345780291494582, 8.6388469707746644028841651985, 9.52842336564057662799885502346, 10.076058036319099529695013058, 12.41665550143515366583430663128, 13.21266617907290684376451227320, 14.15542559231544841741298911086, 15.1668297603816255172154573666, 16.073929199965709493326029446796, 16.86089395048067963814508239365, 17.72866177139291599935294807250, 19.354139928079002583034772850143, 20.11727844725122918125368345375, 21.32873998841015853560353845315, 22.17774592706671973557076166218, 23.04889459017707002775009252687, 24.48121290343645280231053709263, 25.04593206378706768968581300188, 25.9639922252673354265635008620, 26.56751281156011655048574607641, 27.75278980723116640846833854355

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