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Properties

Label 1-173-173.20-r1-0-0
Degree $1$
Conductor $173$
Sign $0.649 + 0.760i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
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  + (0.0729 + 0.997i)2-s + (−0.920 + 0.391i)3-s + (−0.989 + 0.145i)4-s + (−0.288 + 0.957i)5-s + (−0.457 − 0.889i)6-s + (−0.611 − 0.791i)7-s + (−0.217 − 0.976i)8-s + (0.694 − 0.719i)9-s + (−0.976 − 0.217i)10-s + (−0.994 + 0.109i)11-s + (0.853 − 0.520i)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 + (−0.813 + 0.581i)17-s + ⋯
L(s)  = 1  + (0.0729 + 0.997i)2-s + (−0.920 + 0.391i)3-s + (−0.989 + 0.145i)4-s + (−0.288 + 0.957i)5-s + (−0.457 − 0.889i)6-s + (−0.611 − 0.791i)7-s + (−0.217 − 0.976i)8-s + (0.694 − 0.719i)9-s + (−0.976 − 0.217i)10-s + (−0.994 + 0.109i)11-s + (0.853 − 0.520i)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 + (−0.813 + 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6311658505 + 0.2907597912i\)
\(L(\frac12)\) \(\approx\) \(0.6311658505 + 0.2907597912i\)
\(L(1)\) \(\approx\) \(0.5212645228 + 0.3477382738i\)
\(L(1)\) \(\approx\) \(0.5212645228 + 0.3477382738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.0729 + 0.997i)T \)
3 \( 1 + (-0.920 + 0.391i)T \)
5 \( 1 + (-0.288 + 0.957i)T \)
7 \( 1 + (-0.611 - 0.791i)T \)
11 \( 1 + (-0.994 + 0.109i)T \)
13 \( 1 + (0.997 - 0.0729i)T \)
17 \( 1 + (-0.813 + 0.581i)T \)
19 \( 1 + (0.667 - 0.744i)T \)
23 \( 1 + (0.109 - 0.994i)T \)
29 \( 1 + (-0.457 + 0.889i)T \)
31 \( 1 + (-0.391 + 0.920i)T \)
37 \( 1 + (-0.744 - 0.667i)T \)
41 \( 1 + (0.791 - 0.611i)T \)
43 \( 1 + (0.989 + 0.145i)T \)
47 \( 1 + (-0.0365 + 0.999i)T \)
53 \( 1 + (0.946 - 0.322i)T \)
59 \( 1 + (0.967 + 0.252i)T \)
61 \( 1 + (0.813 + 0.581i)T \)
67 \( 1 + (-0.391 - 0.920i)T \)
71 \( 1 + (-0.889 - 0.457i)T \)
73 \( 1 + (-0.905 - 0.424i)T \)
79 \( 1 + (0.999 - 0.0365i)T \)
83 \( 1 + (0.833 - 0.551i)T \)
89 \( 1 + (0.934 + 0.357i)T \)
97 \( 1 + (0.551 - 0.833i)T \)
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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

−27.726297689676972412647229272193, −26.4076114669347461006189276520, −24.97696366889557725392154321099, −23.95084541409316895586264875640, −23.1149341604467719216873149712, −22.33461507481817025164344046065, −21.23784243759934772164658358533, −20.46332580398557528397267069149, −19.17233052839771416455445780776, −18.49189456610369798589411449740, −17.600689331860148172972296545252, −16.297317761847502657481740136699, −15.559782347415679282873754585901, −13.3574765495099819339133574147, −13.11347737283731656260403537012, −11.91858912403599195874997039737, −11.36003391740619357919863724280, −10.0164897244864321086743405985, −8.95649877104205859243737495314, −7.78355819554689841537544585762, −5.859439739888902637279404733830, −5.20290021562857279841725326745, −3.83699073791459913651804911720, −2.178531928092871298020855238239, −0.77586165220971254986023894635, 0.4353871620047399952424502733, 3.36539357277901627449629725548, 4.38577058791231252386182138694, 5.72493291939557842808698047225, 6.711819641893461830283401943724, 7.40267723124346712859641519500, 8.994439650937177464298999372431, 10.428281252174147461925605665, 10.862988229369615725071016120508, 12.609283569696714073569759049174, 13.51878553741375653240088414926, 14.76561534014635071223585963812, 15.90346425118295380818671747330, 16.1610891264117446242012485167, 17.66926468119086354737951033266, 18.10294415667827463144669299380, 19.27542309206035161588846682319, 20.86414118496342959435190333962, 22.14470826788505136027324705021, 22.68710789853415212937446570786, 23.48235088891926961540019460415, 24.13020196915048428703948132974, 25.91550620256938754809553868018, 26.269090639622338261581182429588, 27.079162068034654940086277868683

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