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Properties

Label 1-173-173.56-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: $\chi_{173} (56, \cdot )$
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

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

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