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Properties

Label 1-173-173.138-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.974 + 0.226i$
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  + (−0.181 + 0.983i)2-s + (−0.976 − 0.217i)3-s + (−0.934 − 0.357i)4-s + (0.744 − 0.667i)5-s + (0.391 − 0.920i)6-s + (−0.997 − 0.0729i)7-s + (0.520 − 0.853i)8-s + (0.905 + 0.424i)9-s + (0.520 + 0.853i)10-s + (−0.872 + 0.489i)11-s + (0.833 + 0.551i)12-s + (−0.181 + 0.983i)13-s + (0.252 − 0.967i)14-s + (−0.872 + 0.489i)15-s + (0.744 + 0.667i)16-s + (−0.694 + 0.719i)17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.976 − 0.217i)3-s + (−0.934 − 0.357i)4-s + (0.744 − 0.667i)5-s + (0.391 − 0.920i)6-s + (−0.997 − 0.0729i)7-s + (0.520 − 0.853i)8-s + (0.905 + 0.424i)9-s + (0.520 + 0.853i)10-s + (−0.872 + 0.489i)11-s + (0.833 + 0.551i)12-s + (−0.181 + 0.983i)13-s + (0.252 − 0.967i)14-s + (−0.872 + 0.489i)15-s + (0.744 + 0.667i)16-s + (−0.694 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.974 + 0.226i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.974 + 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03617780510 + 0.3156947620i\)
\(L(\frac12)\) \(\approx\) \(0.03617780510 + 0.3156947620i\)
\(L(1)\) \(\approx\) \(0.4635312896 + 0.2481272147i\)
\(L(1)\) \(\approx\) \(0.4635312896 + 0.2481272147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.181 + 0.983i)T \)
3 \( 1 + (-0.976 - 0.217i)T \)
5 \( 1 + (0.744 - 0.667i)T \)
7 \( 1 + (-0.997 - 0.0729i)T \)
11 \( 1 + (-0.872 + 0.489i)T \)
13 \( 1 + (-0.181 + 0.983i)T \)
17 \( 1 + (-0.694 + 0.719i)T \)
19 \( 1 + (0.252 + 0.967i)T \)
23 \( 1 + (-0.872 - 0.489i)T \)
29 \( 1 + (0.391 + 0.920i)T \)
31 \( 1 + (-0.976 + 0.217i)T \)
37 \( 1 + (0.252 + 0.967i)T \)
41 \( 1 + (-0.997 - 0.0729i)T \)
43 \( 1 + (-0.934 + 0.357i)T \)
47 \( 1 + (0.639 - 0.768i)T \)
53 \( 1 + (-0.0365 + 0.999i)T \)
59 \( 1 + (0.989 + 0.145i)T \)
61 \( 1 + (-0.694 - 0.719i)T \)
67 \( 1 + (-0.976 - 0.217i)T \)
71 \( 1 + (0.391 + 0.920i)T \)
73 \( 1 + (-0.457 + 0.889i)T \)
79 \( 1 + (0.639 + 0.768i)T \)
83 \( 1 + (0.109 + 0.994i)T \)
89 \( 1 + (-0.791 - 0.611i)T \)
97 \( 1 + (0.109 - 0.994i)T \)
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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

−27.07725706698189577663620100975, −26.43707994148127920261413880879, −25.39526096011286648975846965283, −23.80581036440874227438901397175, −22.69087436851044776447717277759, −22.191110626477776052898688076908, −21.49179948370690401811488025807, −20.32234919022946931048045123927, −19.131942968176594414419012412072, −18.082941983269264228155137181617, −17.70451629151816423380556842673, −16.38460097552909135496990793192, −15.32501006074539006718099909307, −13.530244701012157314846730249378, −13.065176948360495846296297527986, −11.77794808614808192704008693976, −10.75531241017944639420526180477, −10.093481771094467701010853359451, −9.2322696065685842740914773340, −7.40144158300704450484620533129, −6.01778042018374017843333344647, −5.08986250776203504345501577439, −3.44451520832727331938863337480, −2.38034542800217392448969466394, −0.30328110664838275597465572220, 1.68117382398646057211054826455, 4.24094911842429801465021336887, 5.30995416891750751109932426659, 6.231968657684267026108098611001, 7.03291803686734924697927765895, 8.48396796226020046847338804144, 9.76555914898567677072743814182, 10.369876753624201895139897880346, 12.321648671751350077793143932749, 12.99701575784803191476763496477, 13.93122808397778137225646846483, 15.479916037897156272548039423160, 16.49533381941809517432078405971, 16.81585343078321997887992881825, 18.04649667713937778617107354236, 18.64622047700652408707162628246, 20.04753422157528440484282429280, 21.65110239696573789917984209842, 22.25009859221766705765462112125, 23.47453471340632750154553492605, 23.944876842018172062022332488489, 25.05255785149441798665853568657, 25.87255312979836138182577416146, 26.81379640825753297280185669200, 28.15160451200204634954155395434

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