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Properties

Label 1-12e3-1728.1723-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.833 + 0.552i$
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  + (0.953 − 0.300i)5-s + (−0.906 − 0.422i)7-s + (0.887 − 0.461i)11-s + (0.537 + 0.843i)13-s + (−0.866 + 0.5i)17-s + (−0.130 + 0.991i)19-s + (0.906 − 0.422i)23-s + (0.819 − 0.573i)25-s + (−0.976 + 0.216i)29-s + (−0.939 − 0.342i)31-s + (−0.991 − 0.130i)35-s + (0.130 + 0.991i)37-s + (0.819 + 0.573i)41-s + (−0.887 + 0.461i)43-s + (−0.342 − 0.939i)47-s + ⋯
L(s)  = 1  + (0.953 − 0.300i)5-s + (−0.906 − 0.422i)7-s + (0.887 − 0.461i)11-s + (0.537 + 0.843i)13-s + (−0.866 + 0.5i)17-s + (−0.130 + 0.991i)19-s + (0.906 − 0.422i)23-s + (0.819 − 0.573i)25-s + (−0.976 + 0.216i)29-s + (−0.939 − 0.342i)31-s + (−0.991 − 0.130i)35-s + (0.130 + 0.991i)37-s + (0.819 + 0.573i)41-s + (−0.887 + 0.461i)43-s + (−0.342 − 0.939i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.833 + 0.552i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.833 + 0.552i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1575312649 + 0.5227302739i\)
\(L(\frac12)\) \(\approx\) \(0.1575312649 + 0.5227302739i\)
\(L(1)\) \(\approx\) \(1.032317407 + 0.007679956196i\)
\(L(1)\) \(\approx\) \(1.032317407 + 0.007679956196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.953 - 0.300i)T \)
7 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (0.887 - 0.461i)T \)
13 \( 1 + (0.537 + 0.843i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.130 + 0.991i)T \)
23 \( 1 + (0.906 - 0.422i)T \)
29 \( 1 + (-0.976 + 0.216i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (0.819 + 0.573i)T \)
43 \( 1 + (-0.887 + 0.461i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.953 + 0.300i)T \)
61 \( 1 + (-0.675 - 0.737i)T \)
67 \( 1 + (0.843 - 0.537i)T \)
71 \( 1 + (0.258 - 0.965i)T \)
73 \( 1 + (-0.258 - 0.965i)T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (-0.216 - 0.976i)T \)
89 \( 1 + (-0.965 + 0.258i)T \)
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

−19.8522931295197326166466759564, −19.058182715032015671714313922125, −18.22814245526643726701443738482, −17.62107033217939017764000767328, −17.01599446674356101442344397403, −16.03477545991765903420862003529, −15.34403077627394483329926412157, −14.65399122038620423084642517598, −13.73408882555768531974979922008, −12.99911481694970681570324996660, −12.63817331001685622844116994536, −11.3130782624512943006935042383, −10.85495452892616618101748065011, −9.70506126519868749528245158596, −9.30329716141572379238273833006, −8.66648718800165268648942168811, −7.165629394663237416477178282796, −6.784051712563212808921864433537, −5.84726919866300914614607288804, −5.247454205382036259612092679573, −4.02928630163021609771255001225, −3.07466185271661996967649404685, −2.358497176022329078447353231780, −1.33234165921770316170308182829, −0.09286987853233814053811377354, 1.2326964851882478257926103198, 1.874964696763334451272961818518, 3.1390801121570528981175101794, 3.91961243999067591330462040989, 4.7862711720784289618651767335, 6.16486841220719031945702397717, 6.21230012608620401639607029409, 7.16974535522026376609893729621, 8.44101659754257210709063125467, 9.17214199579201995208022653808, 9.608486006115656708838873935528, 10.62286301413592550476369451218, 11.24246260412867109908042631128, 12.33983840993250947825751870002, 13.08005462248942776081007050058, 13.5868194829402913855461122955, 14.35452347161905354677556537909, 15.11828795722905343832846678501, 16.31953055173271744390185173712, 16.74912338939424860479640635749, 17.145213043052630996345175357437, 18.39491616216606400147438963019, 18.78208053267018361135283517523, 19.798006436990389547477835642829, 20.313229214377974944734930083028

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