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Properties

Label 1-173-173.59-r1-0-0
Degree $1$
Conductor $173$
Sign $0.756 + 0.653i$
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  + (0.611 − 0.791i)2-s + (0.889 + 0.457i)3-s + (−0.252 − 0.967i)4-s + (−0.489 + 0.872i)5-s + (0.905 − 0.424i)6-s + (0.357 + 0.934i)7-s + (−0.920 − 0.391i)8-s + (0.581 + 0.813i)9-s + (0.391 + 0.920i)10-s + (−0.551 − 0.833i)11-s + (0.217 − 0.976i)12-s + (0.791 + 0.611i)13-s + (0.957 + 0.288i)14-s + (−0.833 + 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.768 + 0.639i)17-s + ⋯
L(s)  = 1  + (0.611 − 0.791i)2-s + (0.889 + 0.457i)3-s + (−0.252 − 0.967i)4-s + (−0.489 + 0.872i)5-s + (0.905 − 0.424i)6-s + (0.357 + 0.934i)7-s + (−0.920 − 0.391i)8-s + (0.581 + 0.813i)9-s + (0.391 + 0.920i)10-s + (−0.551 − 0.833i)11-s + (0.217 − 0.976i)12-s + (0.791 + 0.611i)13-s + (0.957 + 0.288i)14-s + (−0.833 + 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.768 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.756 + 0.653i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ 0.756 + 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.703280017 + 1.006488101i\)
\(L(\frac12)\) \(\approx\) \(2.703280017 + 1.006488101i\)
\(L(1)\) \(\approx\) \(1.756431029 + 0.05283568044i\)
\(L(1)\) \(\approx\) \(1.756431029 + 0.05283568044i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.611 - 0.791i)T \)
3 \( 1 + (0.889 + 0.457i)T \)
5 \( 1 + (-0.489 + 0.872i)T \)
7 \( 1 + (0.357 + 0.934i)T \)
11 \( 1 + (-0.551 - 0.833i)T \)
13 \( 1 + (0.791 + 0.611i)T \)
17 \( 1 + (-0.768 + 0.639i)T \)
19 \( 1 + (0.288 + 0.957i)T \)
23 \( 1 + (0.833 + 0.551i)T \)
29 \( 1 + (0.905 + 0.424i)T \)
31 \( 1 + (0.457 + 0.889i)T \)
37 \( 1 + (-0.957 + 0.288i)T \)
41 \( 1 + (0.934 - 0.357i)T \)
43 \( 1 + (0.252 - 0.967i)T \)
47 \( 1 + (-0.322 - 0.946i)T \)
53 \( 1 + (-0.983 + 0.181i)T \)
59 \( 1 + (-0.667 - 0.744i)T \)
61 \( 1 + (0.768 + 0.639i)T \)
67 \( 1 + (0.457 - 0.889i)T \)
71 \( 1 + (0.424 - 0.905i)T \)
73 \( 1 + (0.694 - 0.719i)T \)
79 \( 1 + (0.946 + 0.322i)T \)
83 \( 1 + (0.520 - 0.853i)T \)
89 \( 1 + (-0.989 + 0.145i)T \)
97 \( 1 + (-0.853 + 0.520i)T \)
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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

−26.73568908671792359243757868758, −26.09399713245503746936678510376, −24.96368589946900084945656590841, −24.353103891087822884117090156443, −23.46126467127260395678980670376, −22.78854848672422129210031052532, −20.88843267046113743302973645478, −20.64726344759103612742446886024, −19.62644789426746295218872935850, −18.0320357539014783431514129984, −17.34671679409242769827043588604, −15.89988486218893767818536206250, −15.37790957730221575345432272217, −14.12592130550467450473909718552, −13.209408379450888790055215236135, −12.71657211885717827789852669656, −11.28920007742207347121544677095, −9.433001388697385719187757342767, −8.33310374492878060884460117552, −7.63689521076251491189483484534, −6.69832730626275895259123490785, −4.88997085586348946749492978033, −4.14741847997813958951251466432, −2.75132752444736570526639086674, −0.784235382513995759885958580568, 1.82608253053893302392791577602, 2.99483521110133534825034128383, 3.75650948318071641508488026928, 5.12436388799386218797933178152, 6.47185336452954667928081247432, 8.21885923920772402197374770568, 9.05458753323502263168212663974, 10.48925436424416156888260022416, 11.10631242036192316551235193146, 12.28969255270204649513642285084, 13.65434063083190314325693092554, 14.3089180347086679312550642577, 15.358381515773464937447190494087, 15.8749069528341858492873574352, 18.15175348873006799284270083513, 18.95043868156649989165619584048, 19.47917782872937964985068716844, 20.86617843855855209763433714668, 21.44241345773737117996186829867, 22.20738370197585055987518952978, 23.375605242579208842626911688977, 24.35489978109589936600950532832, 25.467331942339806823956124286368, 26.65674391751483033438461286207, 27.28427808045051300956444587666

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