Properties

Label 1-2e6-64.27-r1-0-0
Degree $1$
Conductor $64$
Sign $-0.773 + 0.634i$
Analytic cond. $6.87775$
Root an. cond. $6.87775$
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  + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s + 31-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 + 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(6.87775\)
Root analytic conductor: \(6.87775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (1:\ ),\ -0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3723229614 + 1.040572968i\)
\(L(\frac12)\) \(\approx\) \(0.3723229614 + 1.040572968i\)
\(L(1)\) \(\approx\) \(0.8208615855 + 0.4754469941i\)
\(L(1)\) \(\approx\) \(0.8208615855 + 0.4754469941i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 - T \)
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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

−31.461540010168316182684252842976, −30.24576581234102005977883326751, −29.88421103245199012961068251384, −28.28983126063309064690668249704, −26.87177695762773369174360009548, −26.257796189024502865235184580112, −24.58012215846820689016209663460, −23.88266782087409537993539508956, −22.983956288454035091361471991531, −21.34693325299451074333406383851, −19.88963412307776769562618933433, −19.275297011325293772307209873025, −18.037025014089335860059921203540, −16.86796697325819470622295305270, −15.144438824363526066161191568554, −14.23385151179412737265141656124, −12.93604456951215800238322978763, −11.68540737435901605393183145690, −10.54194521729867241087917804791, −8.32133615725299244948501679829, −7.73543961929644199006842159881, −6.32949836024898568851217594100, −4.22360433172716048223421341621, −2.601783086666241230612426113029, −0.543921221630190475450361447905, 2.44656998648640938873141572068, 4.27839388472331745266397975185, 5.14235009109848867310069490849, 7.539438837167444740283954970009, 8.645069874168792518400732034874, 9.86526878837809648384843634027, 11.37938296760468068208422681622, 12.38866896518486383423396326418, 14.32979136291977434343983656295, 15.244416958224685984775249779457, 16.11195199447651831410475064466, 17.49731697949468422731345685312, 19.07089761928346224873064405385, 20.19654589051101055135766280229, 21.07545890993728317358027908860, 22.237575377256032388898222713749, 23.47551822105491698892062536599, 24.73510041532990546370813252904, 25.86243587807944487978069027545, 27.26643525620964289475056851352, 27.650456343804704115611823138241, 28.82584783962982068826165365337, 30.66505022846507803303115787116, 31.525045672629872775720854175077, 32.06719031561098739416560107966

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