Properties

Degree 1
Conductor 173
Sign $-0.496 + 0.868i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.496 + 0.868i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.496 + 0.868i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.496 + 0.868i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (23, \cdot )$
Sato-Tate  :  $\mu(43)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 173,\ (0:\ ),\ -0.496 + 0.868i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9041674362 + 1.558560312i$
$L(\frac12,\chi)$  $\approx$  $0.9041674362 + 1.558560312i$
$L(\chi,1)$  $\approx$  1.237615364 + 1.052520094i
$L(1,\chi)$  $\approx$  1.237615364 + 1.052520094i

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

−27.31210861951344395505555605887, −26.25106016291071918877809400576, −24.8704574514558514112121332594, −24.13565332830868823134347303828, −23.27078777455628093629098354205, −22.65254180211030203972348858610, −21.01575838985266673730195795720, −20.267219920585093064647512658261, −19.790177926699631843100784365140, −18.38571470379573841525373384480, −18.023813878025398515116096826332, −15.86283086367056923413396448281, −14.92484340497471051694182741039, −14.24512302928339969887102043273, −13.08733459200646814344413926803, −12.24631797272818526953517184803, −11.29952110695969418108025822028, −10.27473476987876225559251494163, −8.58410329894432735475585320352, −7.64241490966770138447264362643, −6.509216281715546671155348917995, −4.844567221455783964974600670270, −3.77245159047800246223882958033, −2.57119641455128223947548479127, −1.249518184831906754380032571899, 2.53489696921158640988005686406, 3.8713389159198737441102716492, 4.56656245529462776702257481601, 5.7424798922240562554082695836, 7.47770407682691033070175551921, 8.315027447958055671949151139510, 9.02549342744148312099507506189, 11.18103093602078309153982119147, 11.52618068078354517123572223887, 13.28848913377808261740474314709, 13.998421779913867344843697433780, 15.1998139584046054962413644165, 15.69348958537870428046143903913, 16.50298436107363998212852048585, 17.8503233408375957806566611826, 19.23723537430403229847451885405, 20.37678137684754110174161474205, 21.20433574384595666395630461866, 21.89210451148754508468303580587, 23.09499246458391159243460899962, 24.17634035895193465709230609816, 24.5813274888723686910687379763, 26.10373809481527097369671406476, 26.586619413053565809262735664023, 27.50839814069077693576343404846

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