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Properties

Label 1-73-73.58-r1-0-0
Degree $1$
Conductor $73$
Sign $0.359 + 0.932i$
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  + (0.173 − 0.984i)2-s + (−0.866 − 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.819 − 0.573i)5-s + (−0.642 + 0.766i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.573 − 0.819i)11-s + (0.642 + 0.766i)12-s + (0.0871 + 0.996i)13-s + (−0.996 + 0.0871i)14-s + (0.422 + 0.906i)15-s + (0.766 + 0.642i)16-s + (−0.965 − 0.258i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.866 − 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.819 − 0.573i)5-s + (−0.642 + 0.766i)6-s + (−0.258 − 0.965i)7-s + (−0.5 + 0.866i)8-s + (0.5 + 0.866i)9-s + (−0.707 + 0.707i)10-s + (0.573 − 0.819i)11-s + (0.642 + 0.766i)12-s + (0.0871 + 0.996i)13-s + (−0.996 + 0.0871i)14-s + (0.422 + 0.906i)15-s + (0.766 + 0.642i)16-s + (−0.965 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.359 + 0.932i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.359 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1562785779 - 0.1072193358i\)
\(L(\frac12)\) \(\approx\) \(-0.1562785779 - 0.1072193358i\)
\(L(1)\) \(\approx\) \(0.3544708496 - 0.4317508691i\)
\(L(1)\) \(\approx\) \(0.3544708496 - 0.4317508691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.819 - 0.573i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.573 - 0.819i)T \)
13 \( 1 + (0.0871 + 0.996i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (0.819 - 0.573i)T \)
31 \( 1 + (-0.906 - 0.422i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (-0.996 - 0.0871i)T \)
53 \( 1 + (-0.573 - 0.819i)T \)
59 \( 1 + (0.996 - 0.0871i)T \)
61 \( 1 + (-0.642 - 0.766i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (-0.642 + 0.766i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
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

−32.42243535369290582699409138683, −31.24480702099716851349641018453, −30.302857565084796717600982718724, −28.50721683629413588748804015160, −27.65305741785556139917252648333, −26.82838634526366521949582014720, −25.648529217818420106020723310914, −24.445396321824954014801615925561, −23.2849962821357348403911170058, −22.36855697119831132743789661092, −21.95264484160163994120529191513, −19.957673806513412886924969964864, −18.26751657683277591359895199011, −17.72704742031128239839966487462, −16.11392666280990460558613044401, −15.506783629561343924588950362061, −14.64212849536951542188473595731, −12.66493412435295576495787058900, −11.770188116232679859906490668302, −10.16580891315402102395895246999, −8.801418132556683996555136458061, −7.22214240093327400490226853093, −6.13141075515725932141039280571, −4.86262879264605383122767288304, −3.510473660750680097344518197067, 0.11019876287008105752836913317, 1.37484079025838189425575554783, 3.74905602347774211153149809177, 4.82284319487645654211721060171, 6.55465860440104026904888228860, 8.17283156251027318427198538693, 9.76004228969169413158971806247, 11.32864367640870815025986372289, 11.705998126354856555293171621739, 13.11450908908977922503327351711, 13.959422054180567611980617355891, 16.11555932327129383643527145397, 17.024161614366984782957763504755, 18.38323675791827308635423820904, 19.47201310978296029931078909756, 20.210841841225661309596737741369, 21.74602068261329999587365667058, 22.720000625858197866525946435082, 23.72917622003533184730601726612, 24.32306133945135977355602807502, 26.64754128362677006309938364850, 27.367201059062863409768298470658, 28.58974614113786798209097193183, 29.21285869612228977481985776363, 30.29468505043555003718906277086

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