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Properties

Label 1-173-173.11-r1-0-0
Degree $1$
Conductor $173$
Sign $-0.789 - 0.613i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
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  + (0.667 + 0.744i)2-s + (−0.768 − 0.639i)3-s + (−0.109 + 0.994i)4-s + (0.217 + 0.976i)5-s + (−0.0365 − 0.999i)6-s + (−0.288 − 0.957i)7-s + (−0.813 + 0.581i)8-s + (0.181 + 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.889 + 0.457i)11-s + (0.719 − 0.694i)12-s + (−0.744 + 0.667i)13-s + (0.520 − 0.853i)14-s + (0.457 − 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.0729 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.667 + 0.744i)2-s + (−0.768 − 0.639i)3-s + (−0.109 + 0.994i)4-s + (0.217 + 0.976i)5-s + (−0.0365 − 0.999i)6-s + (−0.288 − 0.957i)7-s + (−0.813 + 0.581i)8-s + (0.181 + 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.889 + 0.457i)11-s + (0.719 − 0.694i)12-s + (−0.744 + 0.667i)13-s + (0.520 − 0.853i)14-s + (0.457 − 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.0729 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1413306886 + 0.4124825624i\)
\(L(\frac12)\) \(\approx\) \(-0.1413306886 + 0.4124825624i\)
\(L(1)\) \(\approx\) \(0.7878091647 + 0.4154292877i\)
\(L(1)\) \(\approx\) \(0.7878091647 + 0.4154292877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.667 + 0.744i)T \)
3 \( 1 + (-0.768 - 0.639i)T \)
5 \( 1 + (0.217 + 0.976i)T \)
7 \( 1 + (-0.288 - 0.957i)T \)
11 \( 1 + (0.889 + 0.457i)T \)
13 \( 1 + (-0.744 + 0.667i)T \)
17 \( 1 + (0.0729 - 0.997i)T \)
19 \( 1 + (-0.853 + 0.520i)T \)
23 \( 1 + (-0.457 - 0.889i)T \)
29 \( 1 + (-0.0365 + 0.999i)T \)
31 \( 1 + (-0.639 - 0.768i)T \)
37 \( 1 + (-0.520 - 0.853i)T \)
41 \( 1 + (-0.957 + 0.288i)T \)
43 \( 1 + (0.109 + 0.994i)T \)
47 \( 1 + (-0.934 - 0.357i)T \)
53 \( 1 + (-0.145 - 0.989i)T \)
59 \( 1 + (-0.551 - 0.833i)T \)
61 \( 1 + (-0.0729 - 0.997i)T \)
67 \( 1 + (-0.639 + 0.768i)T \)
71 \( 1 + (0.999 + 0.0365i)T \)
73 \( 1 + (0.322 + 0.946i)T \)
79 \( 1 + (0.357 + 0.934i)T \)
83 \( 1 + (0.905 + 0.424i)T \)
89 \( 1 + (0.872 + 0.489i)T \)
97 \( 1 + (0.424 + 0.905i)T \)
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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

−27.33654611046155590311487216208, −25.51220014146677372601591257481, −24.40798505819024512480277386981, −23.706422923568136563431189416814, −22.45212972292618376776130012843, −21.77048478774112923435490965834, −21.23765484587276631004669751150, −19.98967244007149682612818644414, −19.18931797741549243281750777438, −17.713746444949146148263681153733, −16.85538698388439960078663606072, −15.57505032236750695559231699885, −14.93174727259390216615112456814, −13.39695204411285291784865741058, −12.30058356083183720542848020544, −11.902603526614331736453371694, −10.56911303105933170979119979550, −9.55007698713881222072229788018, −8.73406344394599119845761606689, −6.28085725644988805673409293042, −5.56760491250065585085657764126, −4.60058734025432041072511715353, −3.429343819136774316778584981747, −1.69875305391977737653799192370, −0.13146218588796704237544986546, 2.13205609761845675421919238125, 3.78059609956684224571695706254, 4.96281895407345943235237910226, 6.53808341741595387897701249137, 6.77962279825882854936884711629, 7.77940091011074499237679760169, 9.658129480494589955605656585521, 10.97340476862537099353015200858, 11.97860618822315986564240807750, 12.969794615207339885600510622562, 14.13020636953195837743553627315, 14.60143888256693990003636491223, 16.27862171856776503175513176994, 16.9340206709053298717384946008, 17.79943681246052723036057838093, 18.80217767816294161210161489302, 20.04703856449241979620863432164, 21.5633479719480756174226718896, 22.50163332520359666716045633768, 22.90706690314004047274911616397, 23.85945082921168204569054373111, 24.82302158846223208481614787711, 25.72570975761923080476690527752, 26.69347043971028794816550668467, 27.60078748274773796939211118253

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