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Properties

Label 1-12e3-1728.475-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: $\chi_{1728} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.999 + 0.00363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0005641793555 - 0.3103205793i\)
\(L(\frac12)\) \(\approx\) \(0.0005641793555 - 0.3103205793i\)
\(L(1)\) \(\approx\) \(0.8605297495 - 0.1281192024i\)
\(L(1)\) \(\approx\) \(0.8605297495 - 0.1281192024i\)

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

−20.39148165307634352803779024651, −19.56558678152274778900201271049, −18.94685609304112919055308058771, −18.40502702695922822126907437855, −17.72878323909281823855052005563, −16.66132033359628365791519801095, −15.85964371588915599352046500923, −15.36994877409713539122934457043, −14.62504423618704159845483767878, −13.81500457769764214369796141015, −13.02341108350893418004489309530, −12.11986481188333255185334768718, −11.36323151782511947776927756869, −10.91975609526060803123371038640, −9.74796323863150144419307388942, −9.23402088956281485355657919, −8.12476348918367109223790268876, −7.54819775195246030789103860303, −6.59137088031468029554877350148, −5.74499318297093115162931199984, −5.20859520410054399272949519720, −3.59126812465796380503404695785, −3.27020379289304622886918536348, −2.440389686037093389151538405795, −1.02144561263160848331032920792, 0.0647859608180050825701159408, 1.19716245070188372735493227447, 1.872386209672958864465875331041, 3.54638682922735628304833474188, 3.90340634708012207482937683259, 4.86084079447561854303406743711, 5.6448114525447039169557640710, 6.99525276843667254602563859243, 7.21310029667387957853099282362, 8.35667267508031983755949318547, 9.06054672332740858630182348902, 9.89633221945027508011137876697, 10.54366798748660110729568060615, 11.6660966578697693413013221512, 12.25684967055249062529462875279, 12.90390843036753867499372786470, 13.74986135456906988774928231374, 14.506418623376399283731099441213, 15.299041062786159000145632003489, 16.387713068458254089121841112407, 16.61194445738828495092120498165, 17.26132721370073116020052774156, 18.37096722638263431667205986803, 19.09462869297147504351333927541, 19.826719641811811104696281702162

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