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Properties

Label 1-173-173.23-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.496 + 0.868i$
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

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.496 + 0.868i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.496 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9041674362 + 1.558560312i\)
\(L(\frac12)\) \(\approx\) \(0.9041674362 + 1.558560312i\)
\(L(1)\) \(\approx\) \(1.237615364 + 1.052520094i\)
\(L(1)\) \(\approx\) \(1.237615364 + 1.052520094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.744 + 0.667i)T \)
3 \( 1 + (0.639 + 0.768i)T \)
5 \( 1 + (-0.976 - 0.217i)T \)
7 \( 1 + (0.957 + 0.288i)T \)
11 \( 1 + (-0.457 - 0.889i)T \)
13 \( 1 + (0.744 + 0.667i)T \)
17 \( 1 + (-0.997 + 0.0729i)T \)
19 \( 1 + (0.520 - 0.853i)T \)
23 \( 1 + (-0.457 + 0.889i)T \)
29 \( 1 + (-0.0365 - 0.999i)T \)
31 \( 1 + (0.639 - 0.768i)T \)
37 \( 1 + (0.520 - 0.853i)T \)
41 \( 1 + (0.957 + 0.288i)T \)
43 \( 1 + (0.109 - 0.994i)T \)
47 \( 1 + (-0.934 + 0.357i)T \)
53 \( 1 + (0.989 + 0.145i)T \)
59 \( 1 + (0.833 + 0.551i)T \)
61 \( 1 + (-0.997 - 0.0729i)T \)
67 \( 1 + (0.639 + 0.768i)T \)
71 \( 1 + (-0.0365 - 0.999i)T \)
73 \( 1 + (-0.322 + 0.946i)T \)
79 \( 1 + (-0.934 - 0.357i)T \)
83 \( 1 + (0.905 - 0.424i)T \)
89 \( 1 + (-0.872 + 0.489i)T \)
97 \( 1 + (0.905 + 0.424i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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