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Properties

Label 1-73-73.50-r0-0-0
Degree $1$
Conductor $73$
Sign $0.866 + 0.499i$
Analytic cond. $0.339010$
Root an. cond. $0.339010$
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  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.642 + 0.766i)5-s + (0.939 − 0.342i)6-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + i·10-s + (−0.642 − 0.766i)11-s + (0.939 + 0.342i)12-s + (0.342 + 0.939i)13-s + (−0.342 − 0.939i)14-s + (0.984 − 0.173i)15-s + (−0.939 + 0.342i)16-s + (0.866 − 0.5i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (0.642 + 0.766i)5-s + (0.939 − 0.342i)6-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + i·10-s + (−0.642 − 0.766i)11-s + (0.939 + 0.342i)12-s + (0.342 + 0.939i)13-s + (−0.342 − 0.939i)14-s + (0.984 − 0.173i)15-s + (−0.939 + 0.342i)16-s + (0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.866 + 0.499i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.866 + 0.499i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473734461 + 0.3948831475i\)
\(L(\frac12)\) \(\approx\) \(1.473734461 + 0.3948831475i\)
\(L(1)\) \(\approx\) \(1.551647747 + 0.3196421372i\)
\(L(1)\) \(\approx\) \(1.551647747 + 0.3196421372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.642 - 0.766i)T \)
13 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (-0.984 - 0.173i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−31.74785180956165311504784364293, −30.57326569260794733653300041989, −29.21541903444166762874732631012, −28.29570976647652491961918580708, −27.60267320279931180838142900023, −25.62801264681211208924419537780, −25.2376295823955243936405561276, −23.55083487407404434622485946726, −22.39529453892611585616133913026, −21.47571522535165092650139120644, −20.58260572955678002568922300647, −19.830436159318564244346800087472, −18.39105009231369064876230049287, −16.5389038397441551052913975829, −15.52359821918744102519233126474, −14.46916673651543134149279173055, −13.08633893238872228419992601328, −12.41952385303678359053148302163, −10.377096483029471763135480367934, −9.86147581882090791580022371345, −8.47296813340431520526306288543, −5.96770297473663230967971914789, −4.98994787045676877363514228881, −3.53256165307784339758601293977, −2.186073269856752345056487293321, 2.49179620914945738291661754596, 3.61880702789749667864060283328, 5.89730028411450639928182310429, 6.71415963908589653454168350428, 7.80304398114552090542923051885, 9.3846947068440106377304623575, 11.24168370831296896099136345581, 12.77922548857729618501450249525, 13.70680007477294715920156035554, 14.2677550200958636166058688642, 15.784967506378339759149254550086, 17.04114396252427714500074857572, 18.3007589652244469170079041636, 19.325823627318938013797202828328, 20.884848764917045812857883728155, 21.89513607301048487070002545681, 23.232903892419980503077837774444, 23.84009666811793710789582244786, 25.26641944161180555130984320761, 25.92800591287997210645625426668, 26.59656823102235053825900457909, 29.06740690054439254731169167693, 29.70600128882853984539486942286, 30.56919598200937350600160280666, 31.7480866232906424175608096997

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