↑

Properties

Label 1-73-73.6-r0-0-0
Degree $1$
Conductor $73$
Sign $0.0842 + 0.996i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.342 + 0.939i)5-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − i·10-s + (−0.342 − 0.939i)11-s + (−0.173 + 0.984i)12-s + (−0.984 + 0.173i)13-s + (0.984 − 0.173i)14-s + (−0.642 + 0.766i)15-s + (0.173 + 0.984i)16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.342 + 0.939i)5-s + (−0.173 − 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − i·10-s + (−0.342 − 0.939i)11-s + (−0.173 + 0.984i)12-s + (−0.984 + 0.173i)13-s + (0.984 − 0.173i)14-s + (−0.642 + 0.766i)15-s + (0.173 + 0.984i)16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0842 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0842 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.0842 + 0.996i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.0842 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4937909492 + 0.4538037687i\)
\(L(\frac12)\) \(\approx\) \(0.4937909492 + 0.4538037687i\)
\(L(1)\) \(\approx\) \(0.6966670718 + 0.2960619660i\)
\(L(1)\) \(\approx\) \(0.6966670718 + 0.2960619660i\)

Euler product

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

−31.42260539823069703728890862306, −29.7479871598904115794916886395, −29.15015845275565906137475363291, −28.19480708063706528238736124234, −26.767668943432197640505510449008, −25.671520989523860894282508060418, −25.02860528997683263679904808067, −24.04372116222496476949077293179, −22.97395203694391800149183578002, −20.75312964738821163354904778175, −20.01536205275473660200002569709, −19.15120732751764837657926201737, −17.82454801846908202747268520131, −17.01238636337064002424897884522, −15.79744718693002265125715929694, −14.359886075754219978162084556195, −13.015341620339968579199257384478, −12.015758740428105347555315211283, −9.92144739679496188143911787196, −9.24031363002895862443165311158, −7.7392906462868802805770142266, −6.93634375695892406120681177988, −5.31939248702284169234885399527, −2.70029384446278966606390953546, −1.04014361102397805656628403071, 2.61184986346890825357065315539, 3.3571790892802770175876669997, 5.82450552639395043052278639763, 7.41160662445579375220396927769, 8.84600937834513299514211361045, 9.91488883294796310067875595820, 10.61587671338486621759386664128, 12.07569967755447765512808329814, 13.84386571377654427617988886283, 15.19285684756899287292943143672, 16.16471678665249494886133811269, 17.28290771903483290278534348162, 18.99973022224861588469387822248, 19.19343967324073185692085898697, 20.85077473615727111393988858315, 21.71342129695094544727520778653, 22.542529447496670423369856711301, 24.77569421699919461552764084193, 25.692241612808431279097406706065, 26.567491950501842324013012891642, 27.14074860936260132460199151651, 28.61797567814149030569405150784, 29.35976688587754693315564690912, 30.63207793816213972245966529055, 31.76725167413487076149389378382

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