Properties

Degree 1
Conductor 173
Sign $0.368 - 0.929i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.368 - 0.929i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (24, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.368 - 0.929i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8033933081 - 0.5457024607i$
$L(\frac12,\chi)$  $\approx$  $0.8033933081 - 0.5457024607i$
$L(\chi,1)$  $\approx$  0.9911835121 - 0.1491675089i
$L(1,\chi)$  $\approx$  0.9911835121 - 0.1491675089i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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