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Properties

Label 1-73-73.61-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

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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: $\chi_{73} (61, \cdot )$
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 \( 1 + (0.5 + 0.866i)T \)
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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

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

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