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Properties

Label 1-73-73.11-r1-0-0
Degree $1$
Conductor $73$
Sign $0.391 + 0.920i$
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  + (0.766 + 0.642i)2-s + (−0.866 − 0.5i)3-s + (0.173 + 0.984i)4-s + (0.0871 − 0.996i)5-s + (−0.342 − 0.939i)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 + (0.996 + 0.0871i)11-s + (0.342 − 0.939i)12-s + (0.906 + 0.422i)13-s + (−0.422 + 0.906i)14-s + (−0.573 + 0.819i)15-s + (−0.939 + 0.342i)16-s + (0.965 + 0.258i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.866 − 0.5i)3-s + (0.173 + 0.984i)4-s + (0.0871 − 0.996i)5-s + (−0.342 − 0.939i)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 + (0.996 + 0.0871i)11-s + (0.342 − 0.939i)12-s + (0.906 + 0.422i)13-s + (−0.422 + 0.906i)14-s + (−0.573 + 0.819i)15-s + (−0.939 + 0.342i)16-s + (0.965 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684522648 + 1.113514735i\)
\(L(\frac12)\) \(\approx\) \(1.684522648 + 1.113514735i\)
\(L(1)\) \(\approx\) \(1.318622469 + 0.4753961279i\)
\(L(1)\) \(\approx\) \(1.318622469 + 0.4753961279i\)

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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.0871 - 0.996i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (0.996 + 0.0871i)T \)
13 \( 1 + (0.906 + 0.422i)T \)
17 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.0871 - 0.996i)T \)
31 \( 1 + (-0.819 + 0.573i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (-0.422 - 0.906i)T \)
53 \( 1 + (-0.996 + 0.0871i)T \)
59 \( 1 + (0.422 - 0.906i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
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

−30.70920635956281037685732260243, −29.97117721567139481069262250067, −29.292175246186521634511287700, −27.803270539558643401902322029729, −27.217124033017350421820776366968, −25.724159687832370074983524214516, −23.98630497470081607401706338909, −23.0588942854325394460642825095, −22.43272038474325083006728002714, −21.37003758582896622888629693264, −20.34442171047202761202566562891, −18.94522547857402546620819389352, −17.76920365500894858839124114087, −16.435982514215018868724939829517, −14.99600574744153525745711623229, −14.15227251690241662310756444700, −12.71921325110918029904152349853, −11.15176776968504288953013023256, −10.86650116222585144333920982891, −9.58450163858833132138461821797, −6.958368242220357569152913689296, −5.96788521981953940676467359209, −4.39128950926279213142532363452, −3.32558344106814000939268908061, −1.02274077614716414882800410591, 1.63609863964832419520739979046, 4.11161835054868922225162717581, 5.48861300212295956884223998311, 6.20049129404326903931062715882, 7.84800013480951226462599484738, 9.06477180367580213526018004842, 11.42798754538006197174092667463, 12.243013787283829814847674398551, 13.09863053717110679774421781064, 14.5005437890877864520020113423, 15.96834329336366739023745786785, 16.78743058770411240411597346811, 17.73298116341156160946983240851, 19.14194489036810862969769532616, 20.98391583563498429976492996818, 21.70107793353077298308198296467, 23.02055097267286173092901816763, 23.78587146697840040855463675806, 24.93486380316927643391261645540, 25.316999793799762609887973284056, 27.447039504807206687088540029118, 28.23740969517132457463445085486, 29.459237675424869954302981466285, 30.53951388617804980838607973200, 31.54981462233794339440627501740

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