Properties

Label 1-73-73.14-r1-0-0
Degree $1$
Conductor $73$
Sign $0.514 + 0.857i$
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  + (−0.939 − 0.342i)2-s + (−0.866 + 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.906 − 0.422i)5-s + (0.984 − 0.173i)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 + (−0.422 + 0.906i)11-s + (−0.984 − 0.173i)12-s + (−0.819 − 0.573i)13-s + (−0.573 + 0.819i)14-s + (0.996 − 0.0871i)15-s + (0.173 + 0.984i)16-s + (0.965 − 0.258i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.866 + 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.906 − 0.422i)5-s + (0.984 − 0.173i)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 + (−0.422 + 0.906i)11-s + (−0.984 − 0.173i)12-s + (−0.819 − 0.573i)13-s + (−0.573 + 0.819i)14-s + (0.996 − 0.0871i)15-s + (0.173 + 0.984i)16-s + (0.965 − 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3822002273 + 0.2163698818i\)
\(L(\frac12)\) \(\approx\) \(0.3822002273 + 0.2163698818i\)
\(L(1)\) \(\approx\) \(0.4555346530 + 0.003475568879i\)
\(L(1)\) \(\approx\) \(0.4555346530 + 0.003475568879i\)

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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.906 - 0.422i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (-0.422 + 0.906i)T \)
13 \( 1 + (-0.819 - 0.573i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (-0.0871 + 0.996i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (-0.573 - 0.819i)T \)
53 \( 1 + (0.422 + 0.906i)T \)
59 \( 1 + (0.573 - 0.819i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.984 - 0.173i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
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

−30.92602844788248533517228497475, −29.73743480225302457597657912367, −28.73985136619221037308569628969, −27.80781104950785198254271147658, −27.02372830747799599941497371217, −25.772320073733826462346283699965, −24.26629235059591790914853140367, −23.9639091308165383492159127950, −22.519271636854570045095508275347, −21.236721883763601172809930574593, −19.21647497501776934408590060773, −18.975083077462424546217217046754, −17.83339713766545014971292809579, −16.60188475555052872961228295557, −15.693262998290222143644442023009, −14.48602205235174249046267445862, −12.3318248700585006345116807528, −11.46360844026377586578705451064, −10.50063077499605133542064035287, −8.69198856821396859430599459280, −7.58541456335798908926051662808, −6.44350509537965722525751645279, −5.140547083382234919647128149101, −2.470478438666098612220732981899, −0.42182536175809736547434959963, 1.030361482373066586131116993743, 3.59329395741865283524728374302, 4.96832699676412529108115062507, 7.0753381721277411294959462182, 7.97953992344843843265886792869, 9.79942413256878344126002049233, 10.549511731650005907428794289991, 11.83559211672088944982659277539, 12.59715085096297004713614443674, 15.01997960716453039098366034505, 16.11741305107254547476586748938, 17.01921763279409809232379416410, 17.85344520000335377353046566426, 19.34288696878132462661468553528, 20.407918016768943866010648809739, 21.194246475869456873640387557651, 22.87725676788494816984824243060, 23.61940262040060800590507028620, 25.083864784398248380248978097873, 26.58974046640849376609286847256, 27.34561732807440630695424901935, 27.93306710370475487406624246538, 29.15000528533764928536347696709, 29.98253544629274070245636864951, 31.33106245896485614916582604207

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