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Properties

Label 1-12e3-1728.403-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.0617 + 0.998i$
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  + (0.216 + 0.976i)5-s + (0.0871 − 0.996i)7-s + (0.843 − 0.537i)11-s + (0.999 + 0.0436i)13-s + (−0.866 − 0.5i)17-s + (0.130 + 0.991i)19-s + (−0.0871 − 0.996i)23-s + (−0.906 + 0.422i)25-s + (−0.675 + 0.737i)29-s + (0.766 + 0.642i)31-s + (0.991 − 0.130i)35-s + (−0.130 + 0.991i)37-s + (−0.906 − 0.422i)41-s + (−0.843 + 0.537i)43-s + (−0.642 − 0.766i)47-s + ⋯
L(s)  = 1  + (0.216 + 0.976i)5-s + (0.0871 − 0.996i)7-s + (0.843 − 0.537i)11-s + (0.999 + 0.0436i)13-s + (−0.866 − 0.5i)17-s + (0.130 + 0.991i)19-s + (−0.0871 − 0.996i)23-s + (−0.906 + 0.422i)25-s + (−0.675 + 0.737i)29-s + (0.766 + 0.642i)31-s + (0.991 − 0.130i)35-s + (−0.130 + 0.991i)37-s + (−0.906 − 0.422i)41-s + (−0.843 + 0.537i)43-s + (−0.642 − 0.766i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.0617 + 0.998i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.0617 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243306677 + 1.322637239i\)
\(L(\frac12)\) \(\approx\) \(1.243306677 + 1.322637239i\)
\(L(1)\) \(\approx\) \(1.115855204 + 0.1545428329i\)
\(L(1)\) \(\approx\) \(1.115855204 + 0.1545428329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.216 + 0.976i)T \)
7 \( 1 + (0.0871 - 0.996i)T \)
11 \( 1 + (0.843 - 0.537i)T \)
13 \( 1 + (0.999 + 0.0436i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.130 + 0.991i)T \)
23 \( 1 + (-0.0871 - 0.996i)T \)
29 \( 1 + (-0.675 + 0.737i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (-0.906 - 0.422i)T \)
43 \( 1 + (-0.843 + 0.537i)T \)
47 \( 1 + (-0.642 - 0.766i)T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (-0.216 - 0.976i)T \)
61 \( 1 + (0.300 + 0.953i)T \)
67 \( 1 + (-0.0436 + 0.999i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.737 + 0.675i)T \)
89 \( 1 + (-0.965 - 0.258i)T \)
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

−19.8222087802876065435100379153, −19.39517322804290404080538186839, −18.29423948805106035983551160764, −17.6499907534446225858606811766, −17.09034938565302340473649693176, −16.145455985892267774396976386, −15.40718484964843907048915115216, −14.99954426806728701396092836319, −13.66125198246240605645558177243, −13.28053223401215947314681491565, −12.40274376714845085298381112863, −11.67115265359941805315786734385, −11.11558059616522244082473408019, −9.76951068495545343959186006795, −9.20721319085257104622139603487, −8.63632179278216150813943593333, −7.86332397517161921207996397783, −6.61684332768985564412647052575, −5.986880342252615203746130911972, −5.12719845828962089508541530568, −4.349927474732030726419581569662, −3.45994352436855033221888106969, −2.11148296151742283266272477527, −1.57910476503252623077208745381, −0.34743111194380639007695411817, 0.98484482656704504108151814965, 1.83612659733255280505499347530, 3.12839976598406656513075767689, 3.686829782983531423224923978636, 4.55340439975540310861007742332, 5.77752846211452267307599581058, 6.6829554431783505357170031926, 6.91203630470165859311793777893, 8.14281886324630033824195323641, 8.79162727654434187534258704829, 9.94532134800988113306980432712, 10.47011098711564357771021298258, 11.24288906317475699829317669452, 11.78027507832284979759048764649, 13.053384546670196811830903094900, 13.76679264931403937014489830139, 14.21543396479770011811764321457, 14.95258741693781921916207656923, 15.94224382214400523169870838047, 16.650451240889945431100342497621, 17.31437990312993113405638679043, 18.275868176808545740907141503643, 18.633122576880422844749128018927, 19.58420255940372153959757548214, 20.320121171472295413252662279558

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