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Properties

Label 1-173-173.140-r0-0-0
Degree $1$
Conductor $173$
Sign $0.391 - 0.920i$
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  + (0.252 − 0.967i)2-s + (−0.581 + 0.813i)3-s + (−0.872 − 0.489i)4-s + (0.520 − 0.853i)5-s + (0.639 + 0.768i)6-s + (0.744 + 0.667i)7-s + (−0.694 + 0.719i)8-s + (−0.322 − 0.946i)9-s + (−0.694 − 0.719i)10-s + (0.391 + 0.920i)11-s + (0.905 − 0.424i)12-s + (0.252 − 0.967i)13-s + (0.833 − 0.551i)14-s + (0.391 + 0.920i)15-s + (0.520 + 0.853i)16-s + (−0.181 − 0.983i)17-s + ⋯
L(s)  = 1  + (0.252 − 0.967i)2-s + (−0.581 + 0.813i)3-s + (−0.872 − 0.489i)4-s + (0.520 − 0.853i)5-s + (0.639 + 0.768i)6-s + (0.744 + 0.667i)7-s + (−0.694 + 0.719i)8-s + (−0.322 − 0.946i)9-s + (−0.694 − 0.719i)10-s + (0.391 + 0.920i)11-s + (0.905 − 0.424i)12-s + (0.252 − 0.967i)13-s + (0.833 − 0.551i)14-s + (0.391 + 0.920i)15-s + (0.520 + 0.853i)16-s + (−0.181 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9466097759 - 0.6262196288i\)
\(L(\frac12)\) \(\approx\) \(0.9466097759 - 0.6262196288i\)
\(L(1)\) \(\approx\) \(0.9864235896 - 0.4170722478i\)
\(L(1)\) \(\approx\) \(0.9864235896 - 0.4170722478i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.252 - 0.967i)T \)
3 \( 1 + (-0.581 + 0.813i)T \)
5 \( 1 + (0.520 - 0.853i)T \)
7 \( 1 + (0.744 + 0.667i)T \)
11 \( 1 + (0.391 + 0.920i)T \)
13 \( 1 + (0.252 - 0.967i)T \)
17 \( 1 + (-0.181 - 0.983i)T \)
19 \( 1 + (0.833 + 0.551i)T \)
23 \( 1 + (0.391 - 0.920i)T \)
29 \( 1 + (0.639 - 0.768i)T \)
31 \( 1 + (-0.581 - 0.813i)T \)
37 \( 1 + (0.833 + 0.551i)T \)
41 \( 1 + (0.744 + 0.667i)T \)
43 \( 1 + (-0.872 + 0.489i)T \)
47 \( 1 + (-0.791 - 0.611i)T \)
53 \( 1 + (-0.934 - 0.357i)T \)
59 \( 1 + (0.109 + 0.994i)T \)
61 \( 1 + (-0.181 + 0.983i)T \)
67 \( 1 + (-0.581 + 0.813i)T \)
71 \( 1 + (0.639 - 0.768i)T \)
73 \( 1 + (-0.0365 + 0.999i)T \)
79 \( 1 + (-0.791 + 0.611i)T \)
83 \( 1 + (-0.457 + 0.889i)T \)
89 \( 1 + (0.957 + 0.288i)T \)
97 \( 1 + (-0.457 - 0.889i)T \)
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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

−27.36434397020261821436472243038, −26.557988869052762735181229180692, −25.63631694817695945968968508004, −24.60324273600397098775156316626, −23.80697469893367338433157715961, −23.23060008825472999686358981725, −21.928038015169543989216450552485, −21.5291686690458626403074587465, −19.52173770177462893147197937227, −18.52958049165690131930061447946, −17.685853915290212350906603222176, −17.064741371837352352268657628167, −16.011211366949040989007047585627, −14.410067748835602386251251286469, −13.979557712261669484856707916, −13.07906586266011938617358884997, −11.57347411043051430094513628493, −10.756882750899557562567876594560, −9.08020809572987928578098746208, −7.79823024376075409899320275907, −6.89167827606281031072286036735, −6.136946659554427388599288625787, −5.0045151880231367381417350299, −3.45511394351090816138625748125, −1.47621787710486586483112545652, 1.14426992083453341256338702158, 2.691098950089285414931984199085, 4.37208803609286575437426370211, 5.0559630413310240294907976379, 5.94817506351105888116943894904, 8.305274176429062695471474673566, 9.41913212911943135420053799354, 10.030036777291886251827560183503, 11.3747707342313306577379716007, 12.07606106996734074866766655038, 13.04371324484229158388014741826, 14.42402679723851845414609565876, 15.30025244057095554620476911812, 16.6211145983090596163564859618, 17.80771952221863168547917655985, 18.195672034225755694208855875307, 20.134641663277079373644912546114, 20.60653369615385447034476870047, 21.353755611293498876260381774590, 22.395545630092617241702823606300, 22.9969316277637104853406001567, 24.35507969476387478007999448498, 25.25774302676934375735361165642, 26.90833500515288570962441920821, 27.6122452716327940126443261104

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