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Properties

Label 1-73-73.65-r0-0-0
Degree $1$
Conductor $73$
Sign $0.239 - 0.970i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.239 - 0.970i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.239 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7656874079 - 0.5995306699i\)
\(L(\frac12)\) \(\approx\) \(0.7656874079 - 0.5995306699i\)
\(L(1)\) \(\approx\) \(0.9245121981 - 0.4790706155i\)
\(L(1)\) \(\approx\) \(0.9245121981 - 0.4790706155i\)

Euler product

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

−32.099032798787435620463272814637, −30.804173508733879047376524438253, −29.71235258495290283776103841420, −28.34614913482558434409375780884, −27.00461593996610892750672417990, −26.131549575870920058331012173924, −25.43314241340119034964920358950, −24.73301501007152174714058628247, −22.98810660433955330350634546869, −22.25397631419724303370326558068, −20.454409677789362549896016459281, −19.36035506379602856189949156666, −18.47665632468659773137332794486, −17.38386996068492324251831270325, −15.793399634671186168448255192481, −15.00898580413770221945996123057, −13.9603840664859350278739005229, −12.91960589919806125765655115866, −10.37134585204489098959629680352, −9.70322470319051802629175582411, −8.42388537586946058792868770137, −7.04795846411878612084115446536, −6.16813496892749252060058107898, −3.97943513142008565933036124605, −2.219104655344750449821747422474, 1.54923301212535917158709157097, 3.08196598809195255876574170965, 4.36873085138529348986222814178, 6.711749861270949985293823503650, 8.69973368001496644826859426901, 9.0038655152073136055451239223, 10.24835574580667601930760961169, 11.93140339447938320658403792317, 13.331645072784931625326162687309, 13.66550808950806051082731416374, 15.8428722907114946086464307781, 16.82487100279184954325719423129, 18.283603797005027896556688919796, 19.54956470268719183564745174164, 19.947417096859177684277441771146, 21.370759018255376997939121547353, 21.87958476914574804032240681764, 23.84995371498973916050326848444, 25.19566715032708781222068798971, 25.98758861295655644468801419129, 26.95384868415264421254446536425, 28.25537364259486958942917711487, 29.16568187053495023760639484818, 30.086679076921554972914625775610, 31.37594646283536642210910192232

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