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Properties

Label 1-2e6-64.19-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: $\chi_{64} (19, \cdot )$
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

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

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