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Properties

Label 1-73-73.53-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.980 + 0.194i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.980 + 0.194i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.980 + 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05889098187 - 0.5984746350i\)
\(L(\frac12)\) \(\approx\) \(-0.05889098187 - 0.5984746350i\)
\(L(1)\) \(\approx\) \(0.7398091698 - 0.4146875989i\)
\(L(1)\) \(\approx\) \(0.7398091698 - 0.4146875989i\)

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

−31.80986078657165386607938536266, −30.74444065364061311085245889122, −29.65207619085348980682002527041, −29.26210198515709258276296967488, −27.19892124963294328469193151804, −26.38400893129936875148566463582, −25.15115004301265975769973436405, −23.90212719923291114329025580917, −23.11965439409703057347046028993, −22.44551679781077328023939278483, −21.320160925861302599178414732178, −19.63307153835664310279060491631, −18.16361873008462094189234316912, −17.282607323997517055124891457875, −16.15722442665181002147605423311, −14.947800094390657491844167453866, −13.61436880765387447417865809398, −12.76738913308966442434913819453, −11.32178458785871140375410154571, −10.34906542203593032445882218063, −7.71026082912633528011083392133, −7.04783720094396163504038888077, −5.873453270259552784845750513283, −4.44571826715134159247872992334, −2.70592218842934112239345678813, 0.24084317049798556040571879378, 2.40623350299029884134848122441, 4.47525091883086052503508103220, 5.15872618726447535317887182891, 6.44883615456423166708711395303, 8.91546993913225702539725761500, 10.06407121618241840300623010486, 11.40492308160944975155130065909, 12.381725524962284183378423914932, 13.14578246458143573857473927659, 15.15964009986707254919213915488, 15.78632179341797054208063022304, 17.18329426297742907912970191768, 18.6120695774961763547233718386, 19.899752695549674506125881243082, 21.200884898979956893910593916, 21.71225789723522197844866104543, 22.91832779499374187295612226044, 23.952052741015550137619063809, 24.76726923264273354100155407081, 26.69692401149675573663576419777, 28.08578918264171113586769604216, 28.59713820998148938143912671725, 29.24740992469502995528843114225, 30.916818956081061406516845149101

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