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Properties

Label 1-73-73.66-r1-0-0
Degree $1$
Conductor $73$
Sign $-0.778 + 0.627i$
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  + (−0.5 − 0.866i)2-s − i·3-s + (−0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.866 + 0.5i)6-s + (−0.707 − 0.707i)7-s + 8-s − 9-s + (−0.707 − 0.707i)10-s + (−0.258 − 0.965i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (−0.258 + 0.965i)14-s + (−0.258 − 0.965i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − i·3-s + (−0.5 + 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.866 + 0.5i)6-s + (−0.707 − 0.707i)7-s + 8-s − 9-s + (−0.707 − 0.707i)10-s + (−0.258 − 0.965i)11-s + (0.866 + 0.5i)12-s + (−0.965 − 0.258i)13-s + (−0.258 + 0.965i)14-s + (−0.258 − 0.965i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(7.84493\)
Root analytic conductor: \(7.84493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (1:\ ),\ -0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2594402422 - 0.7349879369i\)
\(L(\frac12)\) \(\approx\) \(-0.2594402422 - 0.7349879369i\)
\(L(1)\) \(\approx\) \(0.4334494341 - 0.5940722424i\)
\(L(1)\) \(\approx\) \(0.4334494341 - 0.5940722424i\)

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 \( 1 - iT \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.258 - 0.965i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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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

−32.22071582833692849497438931023, −31.40816077528525494946284945068, −29.32928899138088679959583580653, −28.44955595515819388375947829628, −27.54251289499962970356909370232, −26.24076086799771556930879241284, −25.663612561262739130925541945992, −24.78128807562319805377888358063, −23.00452820859742878149367972043, −22.24922259038020706290301089984, −21.117058322806232320264788055383, −19.63619254474296334974462040046, −18.346606566038186614130718955904, −17.21890882198676418472221146540, −16.33883711824105375715829790447, −15.07166094214465867264008182629, −14.440596652151552384693577432348, −12.78853299527779567989954787782, −10.70459451598777971644092120707, −9.58635823632197694080000091433, −9.14022007240134411612905643088, −7.14876016774667262476767928875, −5.79090004079216100266765916618, −4.79234295424188704717668496309, −2.51769090977976853740957241101, 0.43912953109417743555997322296, 1.910786436056288658488203107334, 3.34509726957167870114414032789, 5.62754791557246409233584397155, 7.1830048442011326579160217816, 8.506394244286208179097566781925, 9.81310185734874644440419634760, 10.93506862842109719716041335445, 12.5700489746572704999513024878, 13.14058308306647420132303831694, 14.20322995903301048480076381790, 16.90627350784234364804906739207, 17.11975650441665710096560741728, 18.66307645411419366255085334511, 19.34538469302273567507448041479, 20.50302000049684546799225609903, 21.6591674820136160817231667968, 22.82081654370722268822088406573, 24.1413859799963492478373263734, 25.46481416520837352429153905177, 26.15406056245616371993540990927, 27.57480952985581281350924932454, 29.040659346039378069315066512535, 29.412217422411499164068371913939, 30.07429917263414858181823928673

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