Properties

Label 1-73-73.5-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

Related objects

Downloads

Learn more

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: $\chi_{73} (5, \cdot )$
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 \( 1 + (-0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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