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Properties

Label 1-173-173.37-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.354 - 0.934i$
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  + (0.934 − 0.357i)2-s + (−0.905 + 0.424i)3-s + (0.744 − 0.667i)4-s + (−0.109 − 0.994i)5-s + (−0.694 + 0.719i)6-s + (−0.989 + 0.145i)7-s + (0.457 − 0.889i)8-s + (0.639 − 0.768i)9-s + (−0.457 − 0.889i)10-s + (−0.520 − 0.853i)11-s + (−0.391 + 0.920i)12-s + (−0.934 + 0.357i)13-s + (−0.872 + 0.489i)14-s + (0.520 + 0.853i)15-s + (0.109 − 0.994i)16-s + (0.0365 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.934 − 0.357i)2-s + (−0.905 + 0.424i)3-s + (0.744 − 0.667i)4-s + (−0.109 − 0.994i)5-s + (−0.694 + 0.719i)6-s + (−0.989 + 0.145i)7-s + (0.457 − 0.889i)8-s + (0.639 − 0.768i)9-s + (−0.457 − 0.889i)10-s + (−0.520 − 0.853i)11-s + (−0.391 + 0.920i)12-s + (−0.934 + 0.357i)13-s + (−0.872 + 0.489i)14-s + (0.520 + 0.853i)15-s + (0.109 − 0.994i)16-s + (0.0365 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.354 - 0.934i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.354 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6433360397 - 0.9323828384i\)
\(L(\frac12)\) \(\approx\) \(0.6433360397 - 0.9323828384i\)
\(L(1)\) \(\approx\) \(1.013800508 - 0.5239211284i\)
\(L(1)\) \(\approx\) \(1.013800508 - 0.5239211284i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.934 - 0.357i)T \)
3 \( 1 + (-0.905 + 0.424i)T \)
5 \( 1 + (-0.109 - 0.994i)T \)
7 \( 1 + (-0.989 + 0.145i)T \)
11 \( 1 + (-0.520 - 0.853i)T \)
13 \( 1 + (-0.934 + 0.357i)T \)
17 \( 1 + (0.0365 - 0.999i)T \)
19 \( 1 + (0.872 + 0.489i)T \)
23 \( 1 + (0.520 - 0.853i)T \)
29 \( 1 + (-0.694 - 0.719i)T \)
31 \( 1 + (0.905 + 0.424i)T \)
37 \( 1 + (-0.872 - 0.489i)T \)
41 \( 1 + (0.989 - 0.145i)T \)
43 \( 1 + (0.744 + 0.667i)T \)
47 \( 1 + (-0.181 + 0.983i)T \)
53 \( 1 + (0.997 - 0.0729i)T \)
59 \( 1 + (-0.957 + 0.288i)T \)
61 \( 1 + (0.0365 + 0.999i)T \)
67 \( 1 + (0.905 - 0.424i)T \)
71 \( 1 + (0.694 + 0.719i)T \)
73 \( 1 + (-0.581 + 0.813i)T \)
79 \( 1 + (0.181 + 0.983i)T \)
83 \( 1 + (-0.976 - 0.217i)T \)
89 \( 1 + (0.252 - 0.967i)T \)
97 \( 1 + (0.976 - 0.217i)T \)
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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

−27.97204937846844938061496163539, −26.46317676223638288967823130558, −25.81254229900407006154289246057, −24.702863456868239279156412166478, −23.65558965237103235832504272296, −22.87593262767683227107567623766, −22.347730267990636445414995456432, −21.56320554841490847647337503393, −19.97343752827877072104172785895, −19.04411764861288826699925664170, −17.740921534518856062156726072533, −17.0002316195881613205034872266, −15.721147952575750777153731312448, −15.10128285313286602904021528855, −13.70381069165693787848302305806, −12.804600363061281021110134193118, −12.01520713608952616687508174929, −10.85187585325190845368191914421, −9.95105667348743411222080793395, −7.543265504921977176834178427128, −7.076956768485330106724561847638, −6.03737920552718525898077194222, −4.9918274194029084140820693665, −3.50960570297970676182772228072, −2.27899223992237339065383179506, 0.747599218376632079622707414250, 2.83666971253749997777864954288, 4.185197587438969790583314339146, 5.19870005067047914350975818184, 5.96291485439596996224350763991, 7.26246590070719897080859298480, 9.273866286928867840965361692511, 10.09573343751403265412281786810, 11.384406520926629468701035327663, 12.23585057291673769982122296776, 12.92936926129147375799222117819, 14.0888309179600328706345361541, 15.67317559740377007075924778754, 16.14108515599682059081277380918, 16.94232519342352697617338529006, 18.61368366198570918243412273874, 19.59744281415728329611069954634, 20.7908230042385328660544692842, 21.38372951007018361637750437862, 22.58589153776958368262796427910, 22.964024196166078431670337365057, 24.38539759305719139544374056613, 24.59076786813805584155140874139, 26.36663842354712928136164151183, 27.452474073513608489706806546230

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