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Properties

Label 1-173-173.50-r1-0-0
Degree $1$
Conductor $173$
Sign $-0.884 + 0.466i$
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  + (−0.967 + 0.252i)2-s + (−0.813 + 0.581i)3-s + (0.872 − 0.489i)4-s + (0.853 − 0.520i)5-s + (0.639 − 0.768i)6-s + (−0.667 − 0.744i)7-s + (−0.719 + 0.694i)8-s + (0.322 − 0.946i)9-s + (−0.694 + 0.719i)10-s + (−0.920 − 0.391i)11-s + (−0.424 + 0.905i)12-s + (−0.252 − 0.967i)13-s + (0.833 + 0.551i)14-s + (−0.391 + 0.920i)15-s + (0.520 − 0.853i)16-s + (−0.983 − 0.181i)17-s + ⋯
L(s)  = 1  + (−0.967 + 0.252i)2-s + (−0.813 + 0.581i)3-s + (0.872 − 0.489i)4-s + (0.853 − 0.520i)5-s + (0.639 − 0.768i)6-s + (−0.667 − 0.744i)7-s + (−0.719 + 0.694i)8-s + (0.322 − 0.946i)9-s + (−0.694 + 0.719i)10-s + (−0.920 − 0.391i)11-s + (−0.424 + 0.905i)12-s + (−0.252 − 0.967i)13-s + (0.833 + 0.551i)14-s + (−0.391 + 0.920i)15-s + (0.520 − 0.853i)16-s + (−0.983 − 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ -0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02761996333 + 0.1116478273i\)
\(L(\frac12)\) \(\approx\) \(0.02761996333 + 0.1116478273i\)
\(L(1)\) \(\approx\) \(0.4683344266 + 0.02440481544i\)
\(L(1)\) \(\approx\) \(0.4683344266 + 0.02440481544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.967 + 0.252i)T \)
3 \( 1 + (-0.813 + 0.581i)T \)
5 \( 1 + (0.853 - 0.520i)T \)
7 \( 1 + (-0.667 - 0.744i)T \)
11 \( 1 + (-0.920 - 0.391i)T \)
13 \( 1 + (-0.252 - 0.967i)T \)
17 \( 1 + (-0.983 - 0.181i)T \)
19 \( 1 + (0.551 + 0.833i)T \)
23 \( 1 + (0.391 + 0.920i)T \)
29 \( 1 + (0.639 + 0.768i)T \)
31 \( 1 + (0.581 - 0.813i)T \)
37 \( 1 + (-0.833 + 0.551i)T \)
41 \( 1 + (-0.744 + 0.667i)T \)
43 \( 1 + (-0.872 - 0.489i)T \)
47 \( 1 + (-0.791 + 0.611i)T \)
53 \( 1 + (0.357 + 0.934i)T \)
59 \( 1 + (-0.994 - 0.109i)T \)
61 \( 1 + (0.983 - 0.181i)T \)
67 \( 1 + (0.581 + 0.813i)T \)
71 \( 1 + (0.768 - 0.639i)T \)
73 \( 1 + (0.0365 + 0.999i)T \)
79 \( 1 + (-0.611 + 0.791i)T \)
83 \( 1 + (-0.457 - 0.889i)T \)
89 \( 1 + (-0.957 + 0.288i)T \)
97 \( 1 + (-0.889 - 0.457i)T \)
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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

−26.66334586104059613018791317091, −26.053482585211227653342843259720, −24.98232461582695130884536358206, −24.30006715431951004306017200107, −22.84296809015373584467098818433, −21.87804711604238179647018527066, −21.19509896589077747368237130041, −19.6274731743685350902211124489, −18.763465206289213155528374035171, −18.07138506729244744188402400014, −17.37457144020078734170223974867, −16.25890357311890676005059711043, −15.342239419751487800781071664064, −13.55896098180523586280903722231, −12.61896146528617724030466812576, −11.60956708331742487840796254960, −10.57189245127116045504718623629, −9.71298084491564853209369518690, −8.54189022852103725960709229124, −6.89400207960490253889002440527, −6.55625368712685201368661513755, −5.13179131348408827181046137569, −2.6985046455329913989211442910, −1.933437771759512191743175114949, −0.06830018031018537414144513001, 1.13864487342210672063708689617, 3.06547072208760321087628227712, 5.03004602070016026745554484439, 5.894703034206310816277200178334, 6.96766241380047133807914044885, 8.387595797059208949331798638178, 9.7501572951880703668297894236, 10.15208060686012466370253350167, 11.148443463140699200683611733406, 12.56529243157426925259755038905, 13.68582953117699664176847363596, 15.35585236600268682193121205176, 16.0965291214963081292238121137, 16.963984583206433366056497591674, 17.63035072194346354900064183688, 18.53172466151265731987715717482, 20.05204914444145518323956523001, 20.67872205623516410137826080837, 21.76010768997611803098250566105, 22.942137316110724517089095436653, 23.9142761110521345974716907299, 24.915791377073988417678680084271, 25.9664283935202125086705975013, 26.75850216167125373479630008847, 27.58320208207847606972001462653

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