Properties

Degree 1
Conductor 173
Sign $0.852 + 0.522i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.744 + 0.667i)2-s + (−0.639 + 0.768i)3-s + (0.109 − 0.994i)4-s + (0.976 − 0.217i)5-s + (−0.0365 − 0.999i)6-s + (−0.957 + 0.288i)7-s + (0.581 + 0.813i)8-s + (−0.181 − 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.457 − 0.889i)11-s + (0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.520 − 0.853i)14-s + (−0.457 + 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.997 + 0.0729i)17-s + ⋯
L(s,χ)  = 1  + (−0.744 + 0.667i)2-s + (−0.639 + 0.768i)3-s + (0.109 − 0.994i)4-s + (0.976 − 0.217i)5-s + (−0.0365 − 0.999i)6-s + (−0.957 + 0.288i)7-s + (0.581 + 0.813i)8-s + (−0.181 − 0.983i)9-s + (−0.581 + 0.813i)10-s + (0.457 − 0.889i)11-s + (0.694 + 0.719i)12-s + (0.744 − 0.667i)13-s + (0.520 − 0.853i)14-s + (−0.457 + 0.889i)15-s + (−0.976 − 0.217i)16-s + (0.997 + 0.0729i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.852 + 0.522i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.852 + 0.522i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $0.852 + 0.522i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (15, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ 0.852 + 0.522i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6757651020 + 0.1907402884i$
$L(\frac12,\chi)$  $\approx$  $0.6757651020 + 0.1907402884i$
$L(\chi,1)$  $\approx$  0.6693291521 + 0.2108431852i
$L(1,\chi)$  $\approx$  0.6693291521 + 0.2108431852i

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

−27.84291779684702712598196960868, −26.29235517596703492617619052989, −25.52497122813605842123002661926, −24.972050916669273335065457492796, −23.26049987992618480101037928529, −22.5859998278359205674069135036, −21.54245250904533716358540838790, −20.551153659306856018797579980480, −19.25208546364770055102348887287, −18.72592225874746747494311230572, −17.63653181199793018552048796799, −16.996998773929308375642674359641, −16.11640356891191049092899228829, −14.09437375909312261832083034618, −13.12787899291007937725500911123, −12.35426382246860801866177807455, −11.27941673603659231634218149125, −10.0740172388746463908496133232, −9.48782845754394622305145804440, −7.875143980769015773946298093806, −6.7617717235625950666881087060, −5.93915393431385977863318633047, −3.94487164282629204390460392390, −2.33472018912443835038804417278, −1.29900937431769641867258681788, 0.94998176217678905333203303847, 3.1003310866266826181577215168, 4.95742075468153239761190454748, 6.083595951138035078935559153083, 6.42015178241603711100517148807, 8.4903888552949056174078383361, 9.31250323229240353151692020095, 10.18797366467244705896040157648, 11.00089289152094424768388820316, 12.58148953342966607792872027864, 13.89380540109350689385942710925, 15.00317356618074177440713074754, 16.2147185477267216817698217037, 16.54562392844365021579523363469, 17.6367742231811460194395797961, 18.46388049659951266553984087625, 19.666448087467702229281840404838, 20.860018912675922697532859075351, 21.88989868134233050934384834286, 22.75699773990952941130790261144, 23.80151548986333810980267311358, 24.97372524073779405738753338295, 25.779122334291991437174639351722, 26.46977933097645076644866044909, 27.73515301731349307523606786504

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