↑

Properties

Label 1-173-173.24-r0-0-0
Degree $1$
Conductor $173$
Sign $0.368 - 0.929i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.457 + 0.889i)2-s + (−0.252 − 0.967i)3-s + (−0.581 + 0.813i)4-s + (0.322 − 0.946i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (−0.989 − 0.145i)8-s + (−0.872 + 0.489i)9-s + (0.989 − 0.145i)10-s + (0.997 + 0.0729i)11-s + (0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.0365 − 0.999i)14-s + (−0.997 − 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.109 − 0.994i)17-s + ⋯
L(s)  = 1  + (0.457 + 0.889i)2-s + (−0.252 − 0.967i)3-s + (−0.581 + 0.813i)4-s + (0.322 − 0.946i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (−0.989 − 0.145i)8-s + (−0.872 + 0.489i)9-s + (0.989 − 0.145i)10-s + (0.997 + 0.0729i)11-s + (0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.0365 − 0.999i)14-s + (−0.997 − 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.109 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8033933081 - 0.5457024607i\)
\(L(\frac12)\) \(\approx\) \(0.8033933081 - 0.5457024607i\)
\(L(1)\) \(\approx\) \(0.9911835121 - 0.1491675089i\)
\(L(1)\) \(\approx\) \(0.9911835121 - 0.1491675089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.457 + 0.889i)T \)
3 \( 1 + (-0.252 - 0.967i)T \)
5 \( 1 + (0.322 - 0.946i)T \)
7 \( 1 + (-0.905 - 0.424i)T \)
11 \( 1 + (0.997 + 0.0729i)T \)
13 \( 1 + (-0.457 - 0.889i)T \)
17 \( 1 + (-0.109 - 0.994i)T \)
19 \( 1 + (0.0365 - 0.999i)T \)
23 \( 1 + (-0.997 + 0.0729i)T \)
29 \( 1 + (0.744 + 0.667i)T \)
31 \( 1 + (0.252 - 0.967i)T \)
37 \( 1 + (-0.0365 + 0.999i)T \)
41 \( 1 + (0.905 + 0.424i)T \)
43 \( 1 + (-0.581 - 0.813i)T \)
47 \( 1 + (0.520 + 0.853i)T \)
53 \( 1 + (0.976 + 0.217i)T \)
59 \( 1 + (-0.639 - 0.768i)T \)
61 \( 1 + (-0.109 + 0.994i)T \)
67 \( 1 + (0.252 + 0.967i)T \)
71 \( 1 + (-0.744 - 0.667i)T \)
73 \( 1 + (0.957 - 0.288i)T \)
79 \( 1 + (-0.520 + 0.853i)T \)
83 \( 1 + (-0.791 + 0.611i)T \)
89 \( 1 + (-0.694 - 0.719i)T \)
97 \( 1 + (0.791 + 0.611i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.84581292021202555876318346404, −26.781799144258293229226898643553, −26.11982914544435723801869946542, −24.798072414455781123557207206434, −23.19658824017279863580072164586, −22.62566637898108250210161566712, −21.652423575751825605988163680444, −21.46976962295035513931321570820, −19.80858174942686424920633695001, −19.252722732623218466269707963680, −18.056707295061371143113658437062, −16.87502643738037889627042654753, −15.620617543454338055261859030088, −14.56098933013755329458155217857, −14.0112906879758762182284111314, −12.357968002528743345217888948278, −11.611925394481322586590934694316, −10.402685473176837451563547241935, −9.81650214018480454965539015415, −8.86132594555542569893938117960, −6.44968329814743524142450207895, −5.84153227149792086181908651836, −4.17686641462954560961769456616, −3.42746176544684669488661924126, −2.13352618033472059789491308851, 0.70285976070346945670796486768, 2.814590807483586007272805815990, 4.45377813672900847840360889502, 5.63822413304673083490909606478, 6.58050653909005224732485781586, 7.50698393928442009974012215347, 8.69877560325185021527644302279, 9.72266996625530612025424001429, 11.78300072818614910689652860108, 12.57811280059198277153277369354, 13.4025205110522694443519629800, 14.076580829176333136039251202629, 15.61964436186321607810599740673, 16.64135273220294796591309581778, 17.28492765085658884533422316371, 18.13583358265003187554153784254, 19.630776782030944504439907095, 20.32875997096737196227598635496, 22.04960140159171367604319551996, 22.59439041140659056988168733593, 23.68059602544011658399627166993, 24.45257565789367240936487761737, 25.17817594924667012428259524069, 25.845421199288801535875314271330, 27.266515025681267483467305487629

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