Properties

Label 1-173-173.48-r1-0-0
Degree $1$
Conductor $173$
Sign $0.179 + 0.983i$
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  + (0.424 + 0.905i)2-s + (0.667 − 0.744i)3-s + (−0.639 + 0.768i)4-s + (0.983 + 0.181i)5-s + (0.957 + 0.288i)6-s + (0.719 + 0.694i)7-s + (−0.967 − 0.252i)8-s + (−0.109 − 0.994i)9-s + (0.252 + 0.967i)10-s + (0.611 + 0.791i)11-s + (0.145 + 0.989i)12-s + (−0.905 + 0.424i)13-s + (−0.322 + 0.946i)14-s + (0.791 − 0.611i)15-s + (−0.181 − 0.983i)16-s + (0.551 + 0.833i)17-s + ⋯
L(s)  = 1  + (0.424 + 0.905i)2-s + (0.667 − 0.744i)3-s + (−0.639 + 0.768i)4-s + (0.983 + 0.181i)5-s + (0.957 + 0.288i)6-s + (0.719 + 0.694i)7-s + (−0.967 − 0.252i)8-s + (−0.109 − 0.994i)9-s + (0.252 + 0.967i)10-s + (0.611 + 0.791i)11-s + (0.145 + 0.989i)12-s + (−0.905 + 0.424i)13-s + (−0.322 + 0.946i)14-s + (0.791 − 0.611i)15-s + (−0.181 − 0.983i)16-s + (0.551 + 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.597408626 + 2.166056356i\)
\(L(\frac12)\) \(\approx\) \(2.597408626 + 2.166056356i\)
\(L(1)\) \(\approx\) \(1.737048457 + 0.8383142267i\)
\(L(1)\) \(\approx\) \(1.737048457 + 0.8383142267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.424 + 0.905i)T \)
3 \( 1 + (0.667 - 0.744i)T \)
5 \( 1 + (0.983 + 0.181i)T \)
7 \( 1 + (0.719 + 0.694i)T \)
11 \( 1 + (0.611 + 0.791i)T \)
13 \( 1 + (-0.905 + 0.424i)T \)
17 \( 1 + (0.551 + 0.833i)T \)
19 \( 1 + (0.946 - 0.322i)T \)
23 \( 1 + (-0.791 - 0.611i)T \)
29 \( 1 + (0.957 - 0.288i)T \)
31 \( 1 + (-0.744 + 0.667i)T \)
37 \( 1 + (0.322 + 0.946i)T \)
41 \( 1 + (0.694 - 0.719i)T \)
43 \( 1 + (0.639 + 0.768i)T \)
47 \( 1 + (-0.976 - 0.217i)T \)
53 \( 1 + (0.920 - 0.391i)T \)
59 \( 1 + (-0.999 + 0.0365i)T \)
61 \( 1 + (-0.551 + 0.833i)T \)
67 \( 1 + (-0.744 - 0.667i)T \)
71 \( 1 + (-0.288 - 0.957i)T \)
73 \( 1 + (0.872 - 0.489i)T \)
79 \( 1 + (0.217 + 0.976i)T \)
83 \( 1 + (-0.934 + 0.357i)T \)
89 \( 1 + (0.581 - 0.813i)T \)
97 \( 1 + (0.357 - 0.934i)T \)
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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

−27.22330311227365036890924050715, −26.3798284212409137797857849541, −24.94380017822866195214758544418, −24.29446934614647070012502131349, −22.79107470776165615793199260266, −21.83263712852547084188676246154, −21.28162389468659536795593980788, −20.30138714660940379227645625602, −19.77094809888140760619801870309, −18.37159890840247521946562387425, −17.31467960645359522976779041447, −16.17256992476687220430477709832, −14.48324165348952314535182804197, −14.19430996805576558027471383515, −13.33667721588009737854624308299, −11.865389332462958645140248108660, −10.73403456691363690880646262531, −9.84470639345577055305270059988, −9.14784332331822361405143101909, −7.74088897358508405446648490401, −5.68631449749131146505422727279, −4.83156399478554929607829280934, −3.61653028753539969808004117684, −2.45301655161162351960886796642, −1.104436463510186647684205968177, 1.67369636992754987238544652026, 2.851586827905810686981525290, 4.5615347606169521511365842830, 5.8203227190312917171242621594, 6.792412221311922045985510970942, 7.80876232959385160339283497469, 8.91703490696900014930087058654, 9.77159785575861392102873013079, 11.98419306630008254534880628210, 12.59447717279931076697183416745, 13.92519725468389695109632551195, 14.45647286295386529520611822593, 15.16995602697363704415753934867, 16.79884863164323670108985002616, 17.81962634916351358057277953163, 18.2023576436988945474768698442, 19.61039181031712407863368030060, 20.95307153466608818216483896939, 21.73377526003086054080711191689, 22.69656934345861250648880962720, 24.12943047714287146049040908568, 24.531565507896518555080305468835, 25.43623209006747114669799878225, 26.028699325152402293212279619559, 27.12891727375816570957489796122

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