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Properties

Label 1-73-73.44-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.119 - 0.992i$
Analytic cond. $7.84493$
Root an. cond. $7.84493$
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 + (0.866 − 0.5i)3-s + (0.173 − 0.984i)4-s + (0.996 − 0.0871i)5-s + (0.342 − 0.939i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (0.0871 + 0.996i)11-s + (−0.342 − 0.939i)12-s + (0.422 + 0.906i)13-s + (−0.906 + 0.422i)14-s + (0.819 − 0.573i)15-s + (−0.939 − 0.342i)16-s + (−0.258 − 0.965i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.866 − 0.5i)3-s + (0.173 − 0.984i)4-s + (0.996 − 0.0871i)5-s + (0.342 − 0.939i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (0.0871 + 0.996i)11-s + (−0.342 − 0.939i)12-s + (0.422 + 0.906i)13-s + (−0.906 + 0.422i)14-s + (0.819 − 0.573i)15-s + (−0.939 − 0.342i)16-s + (−0.258 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.119 - 0.992i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.119 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.247386784 - 2.535174239i\)
\(L(\frac12)\) \(\approx\) \(2.247386784 - 2.535174239i\)
\(L(1)\) \(\approx\) \(1.816139676 - 1.219470743i\)
\(L(1)\) \(\approx\) \(1.816139676 - 1.219470743i\)

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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.996 - 0.0871i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (0.0871 + 0.996i)T \)
13 \( 1 + (0.422 + 0.906i)T \)
17 \( 1 + (-0.258 - 0.965i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.996 - 0.0871i)T \)
31 \( 1 + (0.573 - 0.819i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (-0.906 - 0.422i)T \)
53 \( 1 + (-0.0871 + 0.996i)T \)
59 \( 1 + (0.906 - 0.422i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−32.13784988215106865034878957593, −30.60845605391983940678498895161, −29.81673164751073129001594582636, −28.4253372601876225752255226920, −26.63532267693128670369043399832, −25.98601387725004593847180569266, −25.125911783256847239940387088452, −24.261413928526233128526154121997, −22.45209433960435406769727598494, −21.877996500923771216310040422796, −20.88554464344865489842847189986, −19.64844865721363379353477667979, −18.081286424377160072510519224240, −16.59666462523678275771814420001, −15.73422295867633386519818942605, −14.57825561909435207188627754369, −13.506064728245458495773033562684, −12.86050287461033229239715642555, −10.72585883858526306748429843105, −9.29975739130516219430710789812, −8.2342749125155068300019539839, −6.492655709362035465899219454147, −5.431534223733892030986086718977, −3.634965480313390174564604387249, −2.61596545270956173135919226518, 1.49028510890247548230948961265, 2.69728188270154425284173624049, 4.13382623233937435920677921590, 6.00668216660046781244317722311, 7.11190679349688879579795014458, 9.44515916387916465511150521732, 9.807748723247448138752019131000, 11.830730096103345100589001728238, 13.079849394376748200604100240096, 13.67183336813216566865595847660, 14.69879008824049634917755148154, 16.14699153178509073999650293562, 18.02855498041722353868078234060, 19.00646813403550073134800385323, 20.21775261154675605641907102605, 20.84307051477144834067913359344, 22.15947953285928138858971318567, 23.20940378389510652833867502497, 24.51199163772121918376936570628, 25.403195410802674747374839483295, 26.35160723935967416690478574974, 28.21319687125724465147631186061, 29.27343690666353448259351196791, 29.820799686641638047960526908, 31.08453040753920445916040794772

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