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Properties

Label 1-173-173.21-r0-0-0
Degree $1$
Conductor $173$
Sign $0.469 + 0.883i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.469 + 0.883i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.469 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6068831940 + 0.3647952059i\)
\(L(\frac12)\) \(\approx\) \(0.6068831940 + 0.3647952059i\)
\(L(1)\) \(\approx\) \(0.7866158121 + 0.03160428444i\)
\(L(1)\) \(\approx\) \(0.7866158121 + 0.03160428444i\)

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

−26.76768813923210618914444302313, −26.299340730910906198499089740500, −25.545370879805178288581090737141, −24.50738137198485629994936081310, −23.42944042871957193298839148599, −23.09816908240764987166521364248, −21.8059459739385509631533381482, −20.05504492224921207636661792731, −19.21738554247221473103582195238, −18.64287057458267273094375931368, −17.59403463909889715216836380093, −16.48149050759660657440637054275, −15.286065176845284522891568072305, −14.621634431407571931750512156617, −13.42382976340858240738420714845, −12.87242562197746532470357367483, −10.98054444558417030298614435404, −9.97348912986660039969326533645, −8.421338465438280231296017275357, −7.85503452719894185512684637188, −6.67544319494395062694746529040, −6.08057351757118513442401946852, −3.98727565599495992629150746436, −2.891823760768722656745611845985, −0.58793608725657310072772741516, 1.928572037667893249334677604317, 3.22287301906273530837508729839, 4.30505165277231628300588863754, 5.2278246543503876031256181788, 7.51334848992255409827658597502, 8.84856572816748613253153291797, 9.2528483718300921567148304368, 10.3159415650877774080504247506, 11.629658331796562556752400804641, 12.52609004140639597723372217653, 13.48511346056595917119037942137, 14.79777102441241041411802142421, 16.01833402938576509990086036898, 16.63938038110778912143593262025, 18.14848959425617264375275035548, 19.32600185561370415490391992339, 19.84302564996600401252560202910, 20.9513226626505554084423850174, 21.40326260269645765690953094825, 22.65174449653357251265781970151, 23.4727137528153690289844034780, 25.25761790090129859952104613227, 25.774527944845447352739375877577, 27.03893093317763469549632339008, 27.68890908210531040960985148693

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