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Properties

Label 1-73-73.23-r0-0-0
Degree $1$
Conductor $73$
Sign $0.982 + 0.185i$
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  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)6-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s − i·10-s + (0.642 + 0.766i)11-s + (0.939 + 0.342i)12-s + (−0.342 − 0.939i)13-s + (0.342 + 0.939i)14-s + (−0.984 + 0.173i)15-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)6-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s − i·10-s + (0.642 + 0.766i)11-s + (0.939 + 0.342i)12-s + (−0.342 − 0.939i)13-s + (0.342 + 0.939i)14-s + (−0.984 + 0.173i)15-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469390359 + 0.1376909714i\)
\(L(\frac12)\) \(\approx\) \(1.469390359 + 0.1376909714i\)
\(L(1)\) \(\approx\) \(1.530586215 + 0.1415574951i\)
\(L(1)\) \(\approx\) \(1.530586215 + 0.1415574951i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.642 + 0.766i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (0.984 + 0.173i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.939 + 0.342i)T \)
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

−31.45860993492874817352916318516, −30.59893003181681987442960941878, −29.73151863746723025680911484221, −28.202077133967063787349676446614, −27.10811298763006482290451639998, −26.5779598577637709538542623282, −24.754978149405160980520355572983, −23.68201335752902512894214180005, −22.45034660705424416574940637856, −21.67767314635092701910393288266, −20.605086923677670531856595356099, −19.63679497579019151010167969867, −18.67712250989307951654459716806, −16.69292479597368377039651678497, −15.33947703630229113259661142910, −14.42662350686185174941491139732, −13.78719648214704287462047415829, −11.65737384610485593069437347052, −11.06240962102124328757083869034, −9.86548215288096574466714088396, −8.27229715621354259204014157216, −6.490246001827922447727020350999, −4.54916697364048121384278061415, −3.847852758051701504984297105420, −2.321079586211178775627841705360, 2.101255875997211769970918360274, 3.96749180837919700895933914148, 5.3094149890794844550732266991, 6.91261705904364876801524141711, 8.05990074754595800152147831935, 8.82150101686908915765305168154, 11.595773693574363640914302555741, 12.42208169771588627780842991272, 13.347231767867677067391341391760, 14.83363610268173474208375625393, 15.361727700075978418630986343822, 17.166880721626028607381767391307, 17.90688360271363849536236116364, 19.7417292251323696584557571876, 20.44317863134152976245800562731, 21.836871020020956044369189982158, 23.23111237260192542289187376518, 24.12469565532981286441034075773, 24.78001412772623623573610203097, 25.67704942854897657179416520258, 27.151182090828519891647629639295, 28.33298981497173938147950231544, 30.05209735237215581258070347291, 30.62432129734464150599873846098, 31.628025748980617846991363781022

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