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Properties

Label 1-173-173.152-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: $\chi_{173} (152, \cdot )$
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

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

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