Properties

Label 1-173-173.45-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: $\chi_{173} (45, \cdot )$
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

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

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