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Properties

Label 1-73-73.12-r0-0-0
Degree $1$
Conductor $73$
Sign $0.998 - 0.0614i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.998 - 0.0614i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.998 - 0.0614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7663966186 + 0.02355251800i\)
\(L(\frac12)\) \(\approx\) \(0.7663966186 + 0.02355251800i\)
\(L(1)\) \(\approx\) \(0.8355444745 + 0.01032304419i\)
\(L(1)\) \(\approx\) \(0.8355444745 + 0.01032304419i\)

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

−31.177473798505921125510807397501, −30.75703357179634360332786780837, −29.063033753322010827602206475647, −27.89628345894253293498949098781, −27.48132323270061688794898535029, −26.414240729862731355113013603859, −25.27532223408318963126136446540, −24.3199796451943100816197070140, −22.61541764558229745508294599691, −20.9860289824625071745736452199, −20.49542268435335141438948182381, −19.808630535464056796246405237452, −18.13910910971505697179466189708, −16.97920682036604592189945738975, −16.06566371494271130774966229786, −14.988786846512179996140862762929, −13.31128519317833549247447258624, −11.7135921808415444899847549776, −10.70777799134582561155071434624, −9.35143703535484790655853692279, −8.532538162752212672756832304185, −7.3851792137668566839023312199, −4.9055965231168360149940450029, −3.64857587976075883950468799733, −1.62645926424466074053834059009, 1.592299349568072614860886335497, 3.12183658162530161909760570387, 5.9382791408685592145814931549, 7.01932316027153988905098817818, 8.18091306147684789337970562313, 9.025720977661862474603132256170, 11.018924522842500418409529486689, 11.623052497375575861157215652586, 13.70458243269948052348345136398, 14.73465933103154708464530841135, 15.703266507374664503042721716508, 17.45883612446192906736239947576, 18.35107679556160553526771080221, 19.00697220531294836680012893204, 20.06200661970700365591751660876, 21.44824083574992677831899538306, 23.23352407828415065965842334129, 24.25654816166284712562523859428, 25.03023311781580645168626086063, 26.257703871686222846676596353873, 26.90552644354690203727963948102, 28.22703456165460358518663597828, 29.40570579219387568399429757718, 30.44781734412241903372406922114, 31.17060189533640387618595094746

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