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Properties

Label 1-173-173.139-r0-0-0
Degree $1$
Conductor $173$
Sign $0.877 + 0.479i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.084143846 + 0.5328162746i\)
\(L(\frac12)\) \(\approx\) \(2.084143846 + 0.5328162746i\)
\(L(1)\) \(\approx\) \(1.882475146 + 0.2347923137i\)
\(L(1)\) \(\approx\) \(1.882475146 + 0.2347923137i\)

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

−27.143505674196051835639406915249, −25.87062661901859913296239278392, −25.571853963893715130160550384, −24.3009242422463451399862882031, −23.6389904301430620540999306415, −23.10914607514102202028132552969, −21.33199155345690984956962784712, −20.7828670181509890113023998734, −19.82912579906569006326947881208, −18.87602397552410619267609637526, −17.35087503907174073372902630094, −16.28827491970929266188574801146, −15.65673159137444297146801650024, −14.13193453358765086022529424075, −13.53092796807423162284548031784, −12.72828607250641396502947340378, −11.89131101375107400362922771523, −10.199788847814968073570143207468, −8.50775765868001086743035597364, −7.97734667714296021194449309154, −6.66850770047143390001957370576, −5.64135631688962641868327221173, −4.02399565090559039307833202254, −3.25743796118859613444680104484, −1.49413271085852296458146559335, 2.46569068591155400831759234030, 2.982134419542197913924698974156, 4.1790983240091407363757624633, 5.514582795422989159445131356041, 6.71785865248904812197870181869, 8.09390164075534262245930823670, 9.71782128229443505335562057752, 10.36122172049392544410353138829, 11.45134688674344594887868590348, 12.7551984784738015001660897213, 13.72361780810317404870971498352, 14.69854241461091487756673949577, 15.585164417344858001255424341282, 15.978844028236891753773104589248, 18.226705555583318760191254289694, 19.05117527761132684124057478299, 19.95131611738862209174735605241, 20.97632544167063727072012363867, 21.73016502021431106763594031617, 22.67196206595100769514616861374, 23.26411346995602834059749780349, 24.848653524680023239451651535039, 25.685853008554896162686047343323, 26.32056123426430019179825038930, 27.843436669750055298924103483039

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