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Properties

Label 1-12e3-1728.643-r1-0-0
Degree $1$
Conductor $1728$
Sign $0.709 - 0.704i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (0.887 + 0.461i)5-s + (0.422 − 0.906i)7-s + (−0.953 − 0.300i)11-s + (0.976 + 0.216i)13-s + (0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (−0.422 − 0.906i)23-s + (0.573 + 0.819i)25-s + (0.537 − 0.843i)29-s + (−0.939 − 0.342i)31-s + (0.793 − 0.608i)35-s + (−0.608 + 0.793i)37-s + (0.573 − 0.819i)41-s + (0.953 + 0.300i)43-s + (0.342 + 0.939i)47-s + ⋯
L(s)  = 1  + (0.887 + 0.461i)5-s + (0.422 − 0.906i)7-s + (−0.953 − 0.300i)11-s + (0.976 + 0.216i)13-s + (0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (−0.422 − 0.906i)23-s + (0.573 + 0.819i)25-s + (0.537 − 0.843i)29-s + (−0.939 − 0.342i)31-s + (0.793 − 0.608i)35-s + (−0.608 + 0.793i)37-s + (0.573 − 0.819i)41-s + (0.953 + 0.300i)43-s + (0.342 + 0.939i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.849501777 - 1.174237461i\)
\(L(\frac12)\) \(\approx\) \(2.849501777 - 1.174237461i\)
\(L(1)\) \(\approx\) \(1.401480307 - 0.1491438976i\)
\(L(1)\) \(\approx\) \(1.401480307 - 0.1491438976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.887 + 0.461i)T \)
7 \( 1 + (0.422 - 0.906i)T \)
11 \( 1 + (-0.953 - 0.300i)T \)
13 \( 1 + (0.976 + 0.216i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.608 + 0.793i)T \)
23 \( 1 + (-0.422 - 0.906i)T \)
29 \( 1 + (0.537 - 0.843i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (0.573 - 0.819i)T \)
43 \( 1 + (0.953 + 0.300i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (-0.887 - 0.461i)T \)
61 \( 1 + (0.999 + 0.0436i)T \)
67 \( 1 + (-0.216 + 0.976i)T \)
71 \( 1 + (0.965 + 0.258i)T \)
73 \( 1 + (-0.965 + 0.258i)T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.843 - 0.537i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−20.32455766214925974435552470732, −19.52222766680419631289534522583, −18.317743231748635880108697146769, −18.14151658435872232669590762326, −17.45487888468518991813033590791, −16.38545788233582706585980278739, −15.80712192642583278873733556294, −15.070193536921209280780697651543, −14.13762899027276248162405352000, −13.49153204605442398002059197851, −12.68243894420531923325573394143, −12.14947792589838478310924538708, −11.04519409611446529608733138225, −10.415176273286452301008296466558, −9.4624483928876409362499814372, −8.8517825096572846158197069981, −8.09643231686004059357551577567, −7.21794490521820415346636119936, −5.98193962000697130329607292737, −5.492504253917866356242063721077, −4.92793521622961746978703636270, −3.615718476196954172057726349685, −2.63616559375897264319812111211, −1.800228873385205203680217210529, −0.949153889002506587669547896163, 0.62738375298466376892184952924, 1.513370867327943066794424278006, 2.53673486090958360264068080937, 3.43092748327890326563847205146, 4.3424458906681244090739868643, 5.46064198988299425569907424428, 5.96229044256968876485776772303, 6.988854683212423413847425382760, 7.73771684728429398570950350166, 8.47811639322870606433043151483, 9.59719159925381462580562639021, 10.28055305417046548406034789996, 10.78513798209931208745134162652, 11.601112729738537781659479747986, 12.688869025618365102671234706836, 13.47398213293892020464232973557, 14.098365041978142124829742184432, 14.45862355762692616955203213752, 15.725656556924512519760234818853, 16.33928578419094468643238291782, 17.09354736992313417764224591247, 17.89879328633667593699075094458, 18.50028492907436222251844725755, 19.00473794034650039464040679203, 20.31889694102659798626723631605

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