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Properties

Label 1-12e3-1728.1253-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.999 + 0.00363i$
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  + (0.461 + 0.887i)5-s + (0.422 + 0.906i)7-s + (0.300 + 0.953i)11-s + (0.216 + 0.976i)13-s + (−0.866 − 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.422 − 0.906i)23-s + (−0.573 + 0.819i)25-s + (0.843 − 0.537i)29-s + (0.939 − 0.342i)31-s + (−0.608 + 0.793i)35-s + (−0.793 + 0.608i)37-s + (0.573 + 0.819i)41-s + (0.300 + 0.953i)43-s + (0.342 − 0.939i)47-s + ⋯
L(s)  = 1  + (0.461 + 0.887i)5-s + (0.422 + 0.906i)7-s + (0.300 + 0.953i)11-s + (0.216 + 0.976i)13-s + (−0.866 − 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.422 − 0.906i)23-s + (−0.573 + 0.819i)25-s + (0.843 − 0.537i)29-s + (0.939 − 0.342i)31-s + (−0.608 + 0.793i)35-s + (−0.793 + 0.608i)37-s + (0.573 + 0.819i)41-s + (0.300 + 0.953i)43-s + (0.342 − 0.939i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.999 + 0.00363i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.999 + 0.00363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003183336214 + 1.750958677i\)
\(L(\frac12)\) \(\approx\) \(0.003183336214 + 1.750958677i\)
\(L(1)\) \(\approx\) \(1.022734259 + 0.5136688983i\)
\(L(1)\) \(\approx\) \(1.022734259 + 0.5136688983i\)

Euler product

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

−19.60637874193359990324772103749, −19.348483917253957359180373814914, −17.99296050321853405926587898977, −17.35603680421478791559834230377, −17.03687067006348900653940993965, −16.027362508284434737764083551946, −15.476838235424161489093966119202, −14.23133184667332919393521220015, −13.842191650169795076678319937432, −12.99146739065034820079779171318, −12.455799192733247069392247060153, −11.32004041399319841899916207937, −10.649176321374160349128983397405, −10.02162167652792692449779181022, −8.81831943035213086971140430018, −8.47224123445452320692364380446, −7.586634715241249950275530535194, −6.50464410612011057389517806771, −5.76169804784293636391573228638, −4.93046255104815951453904407444, −4.08511104732254419466743779544, −3.27930991392288299800413432764, −1.94415540288428177748316947690, −1.107174654252523887936899752112, −0.315915809590199151466613275595, 1.38183577368485443850344162533, 2.41505414094283300064626673291, 2.68748507694422721920562471711, 4.30848382997090723010448155307, 4.723102296022672961777212582060, 6.01938470919951452528574487982, 6.59817536321283592759387435165, 7.22861907320794584299525758622, 8.45537368689551261158084054739, 9.07956845605108509437525794223, 9.860136610262701177763390667162, 10.712048897239022195748880956155, 11.52006249047516322885082595313, 12.04346496520492796678403438724, 13.09998966060965648353093108423, 13.83970226179036891511109708916, 14.666400603788330934654985972772, 15.13956293471927567076460888154, 15.84555000614519830994152349593, 16.97496188192180811888889062360, 17.68258401328581379357609023806, 18.184013654740384700090908879353, 18.966022710462632491942544607977, 19.53832293505712050729072670373, 20.65803195783730515211709505027

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