↑

Properties

Label 1-2e6-64.59-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

Related objects

Downloads

Learn more

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: $\chi_{64} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (1:\ ),\ 0.773 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.685427568 - 0.6030556268i\)
\(L(\frac12)\) \(\approx\) \(1.685427568 - 0.6030556268i\)
\(L(1)\) \(\approx\) \(1.221600121 - 0.2742820743i\)
\(L(1)\) \(\approx\) \(1.221600121 - 0.2742820743i\)

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 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.58379253472138807383490546623, −31.00847310088615649815352267021, −29.86104456856111108204980774416, −28.50368400362136428091410540808, −27.94652689042179099381782483093, −26.55920035715278868897837801810, −25.70221558439378000324841207226, −24.26804127993926459196906145132, −23.070939169389033373343993813165, −21.9135506964135906894798131587, −20.83667364701691856018863980077, −20.2336018528982918517420903921, −18.061431484590094130790024917163, −17.22111079768168644084856689127, −16.28091302191520286987042906574, −14.77730978284672137861001539736, −13.77162263996464275611047970936, −12.13395465281575016471633304902, −10.70691159189086348879578866512, −9.80733575096933135088106921701, −8.48932436559246659743990461752, −6.47457800574635351897224364312, −5.097938952697780159861740463712, −3.92992400243022209039916765606, −1.5071200731640421859294593998, 1.26421235038631443804410674542, 2.772595012828704120019151091748, 5.41856088473329152163873747774, 6.23520252452283702491419180922, 7.80897195147453939782264754309, 9.177462698674183606323153391659, 10.99086787290983553485361726266, 11.869537588284717904711183926973, 13.50553301414321407586135724101, 14.10622390628662743146899280660, 15.88665217554722871348645278494, 17.45712896427119018991353075956, 18.13816302468336328779773630220, 19.05607241265494866806698771270, 20.7275066126709011622896244053, 21.9096535487864621531048276792, 22.88030335343523529783719000924, 24.42738552148706173982308687239, 24.88780613073061922891451872641, 26.14466398468374970589878207405, 27.70811869589732717816457673553, 28.729857374124470155130012114, 29.78484701874350587006825727975, 30.482273978115226370429629889285, 31.72412167262569761608345839542

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