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Properties

Label 1-73-73.70-r0-0-0
Degree $1$
Conductor $73$
Sign $0.835 + 0.549i$
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 + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + i·7-s + 8-s + 9-s + i·10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s + i·7-s + 8-s + 9-s + i·10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3634052676 + 0.1088154495i\)
\(L(\frac12)\) \(\approx\) \(0.3634052676 + 0.1088154495i\)
\(L(1)\) \(\approx\) \(0.4975545579 - 0.04754349274i\)
\(L(1)\) \(\approx\) \(0.4975545579 - 0.04754349274i\)

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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 + iT \)
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

−31.75591268486242532889607670117, −30.10320967234611649667614896711, −29.268643037651605666410315174, −27.72659948456073073857654787683, −27.18255852406004615616852687229, −26.38395028749233752314146823668, −24.6697782337390974475259061816, −23.849732458159579136883326626378, −22.85809432601192758401448738467, −22.21366601782945139821030804459, −20.00079433439226378220151546729, −19.04514913678649974828389564370, −17.79760986360941965867560329791, −16.90621862680540176633241762365, −15.97921063394072991325160071591, −14.83833770372785792476043327013, −13.47952931040345334401120005572, −11.6436723283997434740220882432, −10.72805553287902299362619030997, −9.43408077137460782326451155658, −7.42807885054769575824129548669, −7.02147709981753888683179453363, −5.37994343923226028325369470933, −4.01316532047686286463052181923, −0.631266945954063859501442755, 1.652654487231475692726238820130, 3.86757530674488778985387704566, 5.11110852365547772255347322085, 7.0541629301486713284230608127, 8.569943113509999250688701538528, 9.76883265106743977249876170764, 11.222576678292564016862689243382, 12.19631130077201137257770283609, 12.57328654819270183231195387237, 14.90176027592689337679870607449, 16.41247267379965200207444899086, 17.151296829020622598240301118078, 18.50679429529993773199952815650, 19.3096946371100196999653672989, 20.579248118492433735081946492600, 21.90237500185971694409361826650, 22.53178671189727644810117938046, 23.904726856236958691441773727994, 25.08759742332981502015799124775, 26.83306005827238116018222462464, 27.63303737182188125150935188633, 28.37806407008272789283606388720, 29.151646535034788535754334509609, 30.507636899704489275372694374671, 31.33860501829332766570581209489

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