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Properties

Label 1-73-73.48-r0-0-0
Degree $1$
Conductor $73$
Sign $-0.928 + 0.371i$
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  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.984 − 0.173i)5-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − i·10-s + (0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s + (−0.642 − 0.766i)14-s + (−0.342 − 0.939i)15-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.984 − 0.173i)5-s + (−0.766 + 0.642i)6-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s − i·10-s + (0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (0.642 + 0.766i)13-s + (−0.642 − 0.766i)14-s + (−0.342 − 0.939i)15-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(73\)
Sign: $-0.928 + 0.371i$
Analytic conductor: \(0.339010\)
Root analytic conductor: \(0.339010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 73,\ (0:\ ),\ -0.928 + 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1610255237 + 0.8350704434i\)
\(L(\frac12)\) \(\approx\) \(0.1610255237 + 0.8350704434i\)
\(L(1)\) \(\approx\) \(0.5961649543 + 0.7498615877i\)
\(L(1)\) \(\approx\) \(0.5961649543 + 0.7498615877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.984 + 0.173i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (0.342 - 0.939i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.642 + 0.766i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−30.816293475921249287052077985008, −30.053327434363477076146785974580, −29.325929163491932273419394286054, −27.91358437480305033927113489011, −26.93914149233832717876575387430, −25.7587878242011741640503498050, −24.35993274301470286674564027467, −22.93747414549045675507715385716, −22.79277791027564704995342294056, −20.726219373064267941067135745780, −19.893603830916183739030603524997, −19.12237476554411299571046648524, −18.29580642605189862323075074136, −16.63260306798926527676816255303, −14.851970406681215404723748798277, −13.83329590070698418655270259842, −12.6250134299873760047208616088, −11.876030537216652028591094854758, −10.44085414396535789094706737001, −8.95632221424495979420098911189, −7.74150926299161185182749546096, −6.20318820424709341598503538729, −3.87926348497794108570691518829, −3.07703637070973490384758155346, −0.991251142245202575360958513907, 3.46121254528494352740723149142, 4.2875987380435045054808662486, 5.92511871408424527986267994648, 7.452292253174317845121889197649, 8.81961643687586543036021658784, 9.502767806878153953164544415554, 11.49345789824953476548866903073, 12.94674752346500643389884477658, 14.356162206126585731954229834026, 15.337299168811905885129552971866, 16.10025012259478842250902321873, 16.981390845855785789764844568377, 18.91062084289174967078563543687, 19.68988623221941470141048581714, 21.33038248058410902319218031909, 22.332433514100235702750103948201, 23.25211397226014495746179871900, 24.52115823083133363635512499322, 25.76607668819732482496391677194, 26.2866026779685527054227819046, 27.72485678520817087340671318645, 28.05787912460182457452769417661, 30.37977035044654978620640677673, 31.41227814820123555081345284068, 32.1111505299062050968237736107

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