Properties

Degree 1
Conductor 173
Sign $-0.132 + 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.997 + 0.0729i)2-s + (−0.391 + 0.920i)3-s + (0.989 + 0.145i)4-s + (−0.957 + 0.288i)5-s + (−0.457 + 0.889i)6-s + (0.791 + 0.611i)7-s + (0.976 + 0.217i)8-s + (−0.694 − 0.719i)9-s + (−0.976 + 0.217i)10-s + (−0.109 + 0.994i)11-s + (−0.520 + 0.853i)12-s + (−0.997 − 0.0729i)13-s + (0.744 + 0.667i)14-s + (0.109 − 0.994i)15-s + (0.957 + 0.288i)16-s + (0.581 − 0.813i)17-s + ⋯
L(s,χ)  = 1  + (0.997 + 0.0729i)2-s + (−0.391 + 0.920i)3-s + (0.989 + 0.145i)4-s + (−0.957 + 0.288i)5-s + (−0.457 + 0.889i)6-s + (0.791 + 0.611i)7-s + (0.976 + 0.217i)8-s + (−0.694 − 0.719i)9-s + (−0.976 + 0.217i)10-s + (−0.109 + 0.994i)11-s + (−0.520 + 0.853i)12-s + (−0.997 − 0.0729i)13-s + (0.744 + 0.667i)14-s + (0.109 − 0.994i)15-s + (0.957 + 0.288i)16-s + (0.581 − 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.132 + 0.991i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.132 + 0.991i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.132 + 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (4, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ -0.132 + 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.055602767 + 1.205674566i$
$L(\frac12,\chi)$  $\approx$  $1.055602767 + 1.205674566i$
$L(\chi,1)$  $\approx$  1.293080148 + 0.7198102516i
$L(1,\chi)$  $\approx$  1.293080148 + 0.7198102516i

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

−27.40843256754406846708334214718, −26.13846810903687933315132289460, −24.70912232086821453184633244870, −23.92305488247266281417871565131, −23.777128622141588847432256362496, −22.6064473257863029505387231708, −21.59843828299762536313893701889, −20.381746415256466356898683218605, −19.53936954841780610545776336787, −18.759469474694743235460125684442, −17.03920760191592226355792042815, −16.582281037939405090128879762374, −15.05126328521790720420867183292, −14.24810442346608853448314438098, −13.100938591156390808904158283799, −12.31685988734528962066984602765, −11.33879183032482148353900849480, −10.71613006169077747028245067393, −8.28012494112264281012700661207, −7.56045877186201324243366223691, −6.47223634501623372771625015164, −5.165538755273796919786849941649, −4.19013724103388562629363579355, −2.66836184901650916691727459674, −1.07340371520192213858461653954, 2.37273157781852200018096978050, 3.72855258752644292763472919247, 4.72268084673825365912424386198, 5.47252923740798942325984574671, 7.045199970857227326333045818243, 8.07125385614445222405488699445, 9.77337403828326299145541348660, 10.92986926117588020883106746596, 11.87968255945233261549224823399, 12.347276212317228477517278412758, 14.28452962320746076236100267042, 15.020021496926739211851128688971, 15.521680180048403433451867130536, 16.64596729760887211544793463635, 17.72632307678361201984895178067, 19.27289465917975163679911614282, 20.377326214062502815091400462611, 21.13174030691261075871957152536, 22.05806119863703838913441431297, 22.9594237020912714136505117804, 23.50092776572929737718851349152, 24.730973575146051341448032909101, 25.70907452728545648392041201258, 27.00819441902924225561672402536, 27.67526921204199240216330055654

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