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Properties

Label 1-173-173.18-r1-0-0
Degree $1$
Conductor $173$
Sign $0.969 - 0.246i$
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  + (−0.424 + 0.905i)2-s + (−0.667 − 0.744i)3-s + (−0.639 − 0.768i)4-s + (−0.983 + 0.181i)5-s + (0.957 − 0.288i)6-s + (−0.719 + 0.694i)7-s + (0.967 − 0.252i)8-s + (−0.109 + 0.994i)9-s + (0.252 − 0.967i)10-s + (−0.611 + 0.791i)11-s + (−0.145 + 0.989i)12-s + (−0.905 − 0.424i)13-s + (−0.322 − 0.946i)14-s + (0.791 + 0.611i)15-s + (−0.181 + 0.983i)16-s + (−0.551 + 0.833i)17-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (−0.667 − 0.744i)3-s + (−0.639 − 0.768i)4-s + (−0.983 + 0.181i)5-s + (0.957 − 0.288i)6-s + (−0.719 + 0.694i)7-s + (0.967 − 0.252i)8-s + (−0.109 + 0.994i)9-s + (0.252 − 0.967i)10-s + (−0.611 + 0.791i)11-s + (−0.145 + 0.989i)12-s + (−0.905 − 0.424i)13-s + (−0.322 − 0.946i)14-s + (0.791 + 0.611i)15-s + (−0.181 + 0.983i)16-s + (−0.551 + 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.969 - 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2065470495 + 0.02581418459i\)
\(L(\frac12)\) \(\approx\) \(0.2065470495 + 0.02581418459i\)
\(L(1)\) \(\approx\) \(0.3616762502 + 0.1249854365i\)
\(L(1)\) \(\approx\) \(0.3616762502 + 0.1249854365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.424 + 0.905i)T \)
3 \( 1 + (-0.667 - 0.744i)T \)
5 \( 1 + (-0.983 + 0.181i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
11 \( 1 + (-0.611 + 0.791i)T \)
13 \( 1 + (-0.905 - 0.424i)T \)
17 \( 1 + (-0.551 + 0.833i)T \)
19 \( 1 + (-0.946 - 0.322i)T \)
23 \( 1 + (-0.791 + 0.611i)T \)
29 \( 1 + (0.957 + 0.288i)T \)
31 \( 1 + (-0.744 - 0.667i)T \)
37 \( 1 + (0.322 - 0.946i)T \)
41 \( 1 + (0.694 + 0.719i)T \)
43 \( 1 + (0.639 - 0.768i)T \)
47 \( 1 + (-0.976 + 0.217i)T \)
53 \( 1 + (-0.920 - 0.391i)T \)
59 \( 1 + (0.999 + 0.0365i)T \)
61 \( 1 + (0.551 + 0.833i)T \)
67 \( 1 + (-0.744 + 0.667i)T \)
71 \( 1 + (0.288 - 0.957i)T \)
73 \( 1 + (0.872 + 0.489i)T \)
79 \( 1 + (-0.217 + 0.976i)T \)
83 \( 1 + (-0.934 - 0.357i)T \)
89 \( 1 + (0.581 + 0.813i)T \)
97 \( 1 + (-0.357 - 0.934i)T \)
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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

−27.09072870208869761814236839360, −26.90498447809111399194173584839, −25.93654670375348649691380453952, −24.049498922024623331835069269377, −23.139134472095825800017657534861, −22.385789020182270205108440189450, −21.43923632789871678908388336841, −20.41570949140917828815666940966, −19.61685494420715273958293415448, −18.69836370318469759966304911168, −17.43402875742847846189386800294, −16.40623813245551483112840779395, −15.97430893309467660626921637599, −14.31777917079251842490382269161, −12.8893928143747662379700976234, −12.01665795571428301196611166304, −11.04955893776154822537245886067, −10.29315003414719006017482345715, −9.23864448419808289258715611241, −8.046599042973744238282356091406, −6.695127484127142710825335307038, −4.815940174740090166024488091432, −4.02081465865004529874957458555, −2.89597591468748893762960443506, −0.52267559580526922081034252680, 0.202352006912742556238895386728, 2.2641381454722737599737767060, 4.39457569379340592979094335982, 5.60660248722634147152889381274, 6.67522035486785825722443562571, 7.5403167776917680079842421432, 8.43617139277837861283027779658, 9.92205521398859509042315283076, 11.01884146440998860098452719074, 12.44237770885900280419286856624, 12.98914246466711324288401092686, 14.69031044975564469513072975156, 15.51741268640749678702882650252, 16.32977964743807814604042966365, 17.53307451233654123161316187202, 18.20201478149493869358776402911, 19.412772318191795127447350932199, 19.62526895048341342104638401627, 21.97129182047989948766838745013, 22.69820276594531639977556244274, 23.57609282510754892674309440320, 24.18345311241831651994721035794, 25.32386866342471445653574427444, 26.03951663217226035561636457538, 27.301153402970639492865225239474

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