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Properties

Label 1-173-173.34-r0-0-0
Degree $1$
Conductor $173$
Sign $0.0248 - 0.999i$
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  + (−0.905 + 0.424i)2-s + (−0.744 − 0.667i)3-s + (0.639 − 0.768i)4-s + (0.181 − 0.983i)5-s + (0.957 + 0.288i)6-s + (0.694 − 0.719i)7-s + (−0.252 + 0.967i)8-s + (0.109 + 0.994i)9-s + (0.252 + 0.967i)10-s + (0.791 − 0.611i)11-s + (−0.989 + 0.145i)12-s + (0.905 − 0.424i)13-s + (−0.322 + 0.946i)14-s + (−0.791 + 0.611i)15-s + (−0.181 − 0.983i)16-s + (−0.833 + 0.551i)17-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + (−0.744 − 0.667i)3-s + (0.639 − 0.768i)4-s + (0.181 − 0.983i)5-s + (0.957 + 0.288i)6-s + (0.694 − 0.719i)7-s + (−0.252 + 0.967i)8-s + (0.109 + 0.994i)9-s + (0.252 + 0.967i)10-s + (0.791 − 0.611i)11-s + (−0.989 + 0.145i)12-s + (0.905 − 0.424i)13-s + (−0.322 + 0.946i)14-s + (−0.791 + 0.611i)15-s + (−0.181 − 0.983i)16-s + (−0.833 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0248 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0248 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.0248 - 0.999i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.0248 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4725821287 - 0.4609755216i\)
\(L(\frac12)\) \(\approx\) \(0.4725821287 - 0.4609755216i\)
\(L(1)\) \(\approx\) \(0.6196589288 - 0.2404385426i\)
\(L(1)\) \(\approx\) \(0.6196589288 - 0.2404385426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.905 + 0.424i)T \)
3 \( 1 + (-0.744 - 0.667i)T \)
5 \( 1 + (0.181 - 0.983i)T \)
7 \( 1 + (0.694 - 0.719i)T \)
11 \( 1 + (0.791 - 0.611i)T \)
13 \( 1 + (0.905 - 0.424i)T \)
17 \( 1 + (-0.833 + 0.551i)T \)
19 \( 1 + (0.322 + 0.946i)T \)
23 \( 1 + (-0.791 - 0.611i)T \)
29 \( 1 + (0.957 - 0.288i)T \)
31 \( 1 + (0.744 - 0.667i)T \)
37 \( 1 + (-0.322 - 0.946i)T \)
41 \( 1 + (-0.694 + 0.719i)T \)
43 \( 1 + (0.639 + 0.768i)T \)
47 \( 1 + (-0.976 - 0.217i)T \)
53 \( 1 + (-0.391 - 0.920i)T \)
59 \( 1 + (0.0365 + 0.999i)T \)
61 \( 1 + (-0.833 - 0.551i)T \)
67 \( 1 + (0.744 + 0.667i)T \)
71 \( 1 + (-0.957 + 0.288i)T \)
73 \( 1 + (-0.872 + 0.489i)T \)
79 \( 1 + (0.976 - 0.217i)T \)
83 \( 1 + (-0.934 + 0.357i)T \)
89 \( 1 + (-0.581 + 0.813i)T \)
97 \( 1 + (0.934 + 0.357i)T \)
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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

−27.72895156985514335957693080660, −26.98744510709542264079545739430, −26.06529020080325077804127969930, −25.20175383842302783525074179700, −23.88479607727722371372963601343, −22.473467264352121047301684628880, −21.875297714734765120271141292469, −21.056847626355566943730884483836, −19.95705222862549288860915225128, −18.66305362242916797960734396950, −17.80960359993118674793614112289, −17.42248460682622771031455906438, −15.83750158226089770582366248479, −15.34940874408092462184835724717, −13.91703111907796745290193695316, −12.02150036674444731708165184449, −11.488958291760902248390539122434, −10.644598057865075219486597758829, −9.56509683564572796492892392770, −8.69403472711317005452527933719, −7.029704677424688293761983086970, −6.22253099280077135492491842129, −4.53985671235771987452515090473, −3.14336125211035365310880183501, −1.68298870387166808489139871453, 0.8806142589262055036395059658, 1.72869398551289019305730042066, 4.387081227320686413616197503450, 5.758269147470049169850883602591, 6.50539945905378182228857100473, 8.007584984748246019531465940872, 8.47893309021534426853444865033, 10.06343762076228319263302295951, 11.08907798466126874718298913194, 11.95485641151760178782070650543, 13.33817640944539006564492995583, 14.28179315387364987504581935644, 15.96437104689760962020023123445, 16.64689087859739484389558401557, 17.47712409481024410715684839035, 18.09656096668151753346701219410, 19.369016418577974027156100154242, 20.19209676768984029209120809454, 21.2213612435256145347951971903, 22.81799297697627463273191623969, 23.76400096515885494509293401378, 24.52547684595606454496625749384, 24.97401752029958192880013114008, 26.40322530591995746281525994226, 27.493397521535206968663808265859

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