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Properties

Label 1-73-73.29-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.392 - 0.919i$
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 + (−0.996 + 0.0871i)5-s + (0.342 − 0.939i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.0871 − 0.996i)11-s + (−0.342 − 0.939i)12-s + (−0.422 − 0.906i)13-s + (0.906 − 0.422i)14-s + (−0.819 + 0.573i)15-s + (−0.939 − 0.342i)16-s + (0.258 + 0.965i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.866 − 0.5i)3-s + (0.173 − 0.984i)4-s + (−0.996 + 0.0871i)5-s + (0.342 − 0.939i)6-s + (0.965 + 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 + 0.707i)10-s + (−0.0871 − 0.996i)11-s + (−0.342 − 0.939i)12-s + (−0.422 − 0.906i)13-s + (0.906 − 0.422i)14-s + (−0.819 + 0.573i)15-s + (−0.939 − 0.342i)16-s + (0.258 + 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.392 - 0.919i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.392 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.601094651 - 2.422748119i\)
\(L(\frac12)\) \(\approx\) \(1.601094651 - 2.422748119i\)
\(L(1)\) \(\approx\) \(1.532475204 - 1.164724099i\)
\(L(1)\) \(\approx\) \(1.532475204 - 1.164724099i\)

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 \( 1 + (-0.996 + 0.0871i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.0871 - 0.996i)T \)
13 \( 1 + (-0.422 - 0.906i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (0.996 + 0.0871i)T \)
31 \( 1 + (-0.573 + 0.819i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (0.906 + 0.422i)T \)
53 \( 1 + (0.0871 - 0.996i)T \)
59 \( 1 + (-0.906 + 0.422i)T \)
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

−31.396585292494160609379743519, −31.04716113189901306940636416530, −30.10913899474298189853423223732, −28.063019856505419466697252597281, −26.89847271323700364860021623680, −26.297594400303712289389756977460, −24.90700166943242921951201026804, −24.08730315027017069239051071434, −22.999019023509705866462246590047, −21.76106290452692573658557761297, −20.6161767150682247429789088793, −19.94034720919980941956176634563, −18.1866824808850326812810860325, −16.61067550617328123422959949125, −15.64246971222882585181728477451, −14.68558047175549057590042736615, −13.93325932768995085891549235138, −12.34618761660566027217300703134, −11.21395514773565394173097815112, −9.25908608636829647771923597765, −7.88158428795211459109065008434, −7.23736941778465286587813243848, −4.76126831055749193976093755436, −4.25572670513316607449122634286, −2.55320723501593738857049072630, 1.18619561296990584220469117315, 2.90101098040174059601408253886, 3.99097143029108079011619999724, 5.68845107277724856491580703380, 7.56666115141226161353087758530, 8.56430810864178407032571455338, 10.41238214544894745090413652005, 11.72368375536252606672238282962, 12.589410371506603410550051593431, 13.99210042828440240013051545198, 14.8020740497446023635877977994, 15.78591180232429286166907611019, 18.05562859965764745541276660275, 19.11782282663198940372168963359, 19.9052668424800978838040059119, 20.90531682096816177688469705161, 22.00575768790569880801841948914, 23.56081646865474931847548314811, 24.11620646183163984879250580324, 25.14634447615964361724145211477, 26.93810540847405114805970128670, 27.58834010579661889930950420972, 29.16135292671961016351857383851, 30.27846327882159220273415483589, 30.854086982167622428659403468011

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