Properties

Label 1-12e3-1728.499-r1-0-0
Degree $1$
Conductor $1728$
Sign $0.503 - 0.864i$
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  + (0.537 + 0.843i)5-s + (0.996 − 0.0871i)7-s + (0.976 − 0.216i)11-s + (−0.737 − 0.675i)13-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (−0.996 − 0.0871i)23-s + (−0.422 + 0.906i)25-s + (0.0436 − 0.999i)29-s + (0.766 − 0.642i)31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (−0.422 − 0.906i)41-s + (−0.976 + 0.216i)43-s + (0.642 − 0.766i)47-s + ⋯
L(s)  = 1  + (0.537 + 0.843i)5-s + (0.996 − 0.0871i)7-s + (0.976 − 0.216i)11-s + (−0.737 − 0.675i)13-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (−0.996 − 0.0871i)23-s + (−0.422 + 0.906i)25-s + (0.0436 − 0.999i)29-s + (0.766 − 0.642i)31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (−0.422 − 0.906i)41-s + (−0.976 + 0.216i)43-s + (0.642 − 0.766i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.503 - 0.864i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ 0.503 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.227189090 - 1.280475017i\)
\(L(\frac12)\) \(\approx\) \(2.227189090 - 1.280475017i\)
\(L(1)\) \(\approx\) \(1.322975096 - 0.05031450786i\)
\(L(1)\) \(\approx\) \(1.322975096 - 0.05031450786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.537 + 0.843i)T \)
7 \( 1 + (0.996 - 0.0871i)T \)
11 \( 1 + (0.976 - 0.216i)T \)
13 \( 1 + (-0.737 - 0.675i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.793 + 0.608i)T \)
23 \( 1 + (-0.996 - 0.0871i)T \)
29 \( 1 + (0.0436 - 0.999i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (-0.422 - 0.906i)T \)
43 \( 1 + (-0.976 + 0.216i)T \)
47 \( 1 + (0.642 - 0.766i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.537 - 0.843i)T \)
61 \( 1 + (0.887 - 0.461i)T \)
67 \( 1 + (0.675 - 0.737i)T \)
71 \( 1 + (-0.965 - 0.258i)T \)
73 \( 1 + (0.965 - 0.258i)T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (-0.999 - 0.0436i)T \)
89 \( 1 + (0.258 + 0.965i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−20.07174843792219992517682624297, −19.76298994668502058844118594118, −18.716801842816692278393384016364, −17.83984747752117296495908979449, −17.1697147122286766959584899475, −16.808378077175483939386294198225, −15.88661959024232931644942461490, −14.75102045469578210231835641951, −14.38794455548200297575177676052, −13.630644736154440838846075713055, −12.55450564170705093131164479216, −12.092445346398874290614573974690, −11.34214427134495170020709711887, −10.29360308006164017216520533061, −9.56724803998073605438589975783, −8.776290649241035757025709984362, −8.19695776033150769477016104216, −7.18691632703716637225041553315, −6.27653382277456592062019019969, −5.42402893556217047620978983456, −4.57305311816057184481730709598, −4.088882294580344187287300446129, −2.59273834364539711549216774791, −1.65610916453047486245649373181, −1.10976932910813633407100664365, 0.45509288068066836676333193135, 1.672496449817674202821850339429, 2.37468414551630426288510383187, 3.428605932168266448591334256153, 4.30402640769557950430123604634, 5.32214615477387280051547885560, 6.07618023701069597852618331776, 6.85414722655574006002887950298, 7.844169147174707298213406615581, 8.31309562064369523597134279809, 9.67615529841597319478609977975, 10.0208362513235143183617494066, 10.947181435586421040342579469569, 11.72493246801702358288568751621, 12.28138964037178877701014949100, 13.55682878122980199216196044252, 14.07036199601515996992002648096, 14.811619731447108312613681845666, 15.16364045261492154737214597102, 16.47894628408381602054302378158, 17.263347938674298322810566905861, 17.59180252904612992312306396032, 18.61839348459557996671291558680, 19.01819709130511703295706537356, 20.083094988500139063134774942323

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