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Properties

Label 1-173-173.89-r0-0-0
Degree $1$
Conductor $173$
Sign $0.757 + 0.652i$
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  + (0.181 + 0.983i)2-s + (0.976 − 0.217i)3-s + (−0.934 + 0.357i)4-s + (−0.744 − 0.667i)5-s + (0.391 + 0.920i)6-s + (0.997 − 0.0729i)7-s + (−0.520 − 0.853i)8-s + (0.905 − 0.424i)9-s + (0.520 − 0.853i)10-s + (0.872 + 0.489i)11-s + (−0.833 + 0.551i)12-s + (−0.181 − 0.983i)13-s + (0.252 + 0.967i)14-s + (−0.872 − 0.489i)15-s + (0.744 − 0.667i)16-s + (0.694 + 0.719i)17-s + ⋯
L(s)  = 1  + (0.181 + 0.983i)2-s + (0.976 − 0.217i)3-s + (−0.934 + 0.357i)4-s + (−0.744 − 0.667i)5-s + (0.391 + 0.920i)6-s + (0.997 − 0.0729i)7-s + (−0.520 − 0.853i)8-s + (0.905 − 0.424i)9-s + (0.520 − 0.853i)10-s + (0.872 + 0.489i)11-s + (−0.833 + 0.551i)12-s + (−0.181 − 0.983i)13-s + (0.252 + 0.967i)14-s + (−0.872 − 0.489i)15-s + (0.744 − 0.667i)16-s + (0.694 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.757 + 0.652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.447906219 + 0.5376628346i\)
\(L(\frac12)\) \(\approx\) \(1.447906219 + 0.5376628346i\)
\(L(1)\) \(\approx\) \(1.330475090 + 0.4249463392i\)
\(L(1)\) \(\approx\) \(1.330475090 + 0.4249463392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.181 + 0.983i)T \)
3 \( 1 + (0.976 - 0.217i)T \)
5 \( 1 + (-0.744 - 0.667i)T \)
7 \( 1 + (0.997 - 0.0729i)T \)
11 \( 1 + (0.872 + 0.489i)T \)
13 \( 1 + (-0.181 - 0.983i)T \)
17 \( 1 + (0.694 + 0.719i)T \)
19 \( 1 + (-0.252 + 0.967i)T \)
23 \( 1 + (-0.872 + 0.489i)T \)
29 \( 1 + (0.391 - 0.920i)T \)
31 \( 1 + (-0.976 - 0.217i)T \)
37 \( 1 + (0.252 - 0.967i)T \)
41 \( 1 + (-0.997 + 0.0729i)T \)
43 \( 1 + (-0.934 - 0.357i)T \)
47 \( 1 + (0.639 + 0.768i)T \)
53 \( 1 + (0.0365 + 0.999i)T \)
59 \( 1 + (-0.989 + 0.145i)T \)
61 \( 1 + (0.694 - 0.719i)T \)
67 \( 1 + (-0.976 + 0.217i)T \)
71 \( 1 + (-0.391 + 0.920i)T \)
73 \( 1 + (-0.457 - 0.889i)T \)
79 \( 1 + (-0.639 + 0.768i)T \)
83 \( 1 + (0.109 - 0.994i)T \)
89 \( 1 + (-0.791 + 0.611i)T \)
97 \( 1 + (-0.109 - 0.994i)T \)
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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

−27.22194297274355934913542432381, −26.86718923144778272426949604284, −25.704663717492744470040672373597, −24.23646130321987669928856067733, −23.59775834418699404083946071042, −22.042491270228463885702686049235, −21.658236880319571448022152179837, −20.428109031033394148704819317077, −19.74878079221972514165106438956, −18.83548253660354502879490409752, −18.153137205403863855104268073996, −16.495381849271251362988992043430, −14.99069471414865314351259111881, −14.38181570488290170905480230327, −13.70027111522886796222191914246, −12.05419345236545453057403631729, −11.38498718248426574717374413140, −10.29970483617021536472338213592, −9.051456926281428091864506239410, −8.26323112151802680358950365997, −6.935693065095675336265806565086, −4.84225846869504615791867263205, −3.929938371860256418192872872776, −2.89306187617998308702552562153, −1.633104585744823604412468570427, 1.4678066292958898006600517486, 3.65118478073352385113364287688, 4.379870289947590343139728973214, 5.76959899297263994907533188155, 7.465308432783506263223707689917, 7.967070104293504542232839877334, 8.79173161126063024057877137607, 10.01823559334080129217059028610, 12.026577833912000269825777916857, 12.736958299591146382841871275115, 14.01096801650262312600019954796, 14.83151457514891169541738869570, 15.4396706178172217229583202005, 16.73534361899978584183702829809, 17.64840635859540036361503600666, 18.776721743234870886881067337180, 19.88013741152663924156101439951, 20.7164632646988261467058532393, 21.825172410567031517303495335296, 23.24071806370973698274319219998, 23.8984614846269422792962265122, 24.9228064681963084934237722983, 25.25322224890315270705071299560, 26.628969513918348236999070168128, 27.44358754643825365418891253298

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