↑

Properties

Label 1-2e6-64.3-r1-0-0
Degree $1$
Conductor $64$
Sign $0.0980 + 0.995i$
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  + (0.923 + 0.382i)3-s + (0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s + i·15-s + i·17-s + (0.382 − 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + 31-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (0.382 + 0.923i)5-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s + i·15-s + i·17-s + (0.382 − 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.0980 + 0.995i$
Analytic conductor: \(6.87775\)
Root analytic conductor: \(6.87775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 64,\ (1:\ ),\ 0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.508595342 + 1.367311118i\)
\(L(\frac12)\) \(\approx\) \(1.508595342 + 1.367311118i\)
\(L(1)\) \(\approx\) \(1.310557036 + 0.5675337236i\)
\(L(1)\) \(\approx\) \(1.310557036 + 0.5675337236i\)

Euler product

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

−31.786202499817022725877226734968, −30.83675594456118217442519334403, −29.30169431591452504800969376648, −28.888681601801354661042469853009, −27.03627312050940449186834167091, −26.12701370724849164699318483601, −25.09159316820087310145964321011, −24.12282188205187252687780277029, −23.03947673333790643954679622985, −21.12061729205036982878053906932, −20.58783338392862214682210233791, −19.35470950447233427923506138959, −18.31196409257002284236776536962, −16.6893263648400450159939480292, −15.79067376717891400133641773722, −13.930661705734714199917771989195, −13.40548901458927259499367293092, −12.18376646108015996815217706518, −10.14422412234955860546745033522, −9.0681113688158560199761235790, −7.87200207148676806680865807181, −6.42039258680156058555919953393, −4.51131642372121035116650666100, −2.87858637385649451160366883423, −1.03728055455584695588095267179, 2.42305761187148927580868692741, 3.36660898410350273293022861589, 5.43856483607659250885913455163, 7.05600955327449719471047577285, 8.49224210491056672508705388910, 9.83220497305453173783201835770, 10.71942306373169552743473735424, 12.76678654385591680157049379584, 13.75337183836781777976639592033, 15.257557715130958711433990061960, 15.611964149738099611944842689664, 17.67566359556238167733221028343, 18.79452622134416107292740321214, 19.74343267539305680085142599599, 21.15876443790723831536648101860, 21.97657407076135115525549731162, 23.14764198541931936095696364975, 24.89699189738831058215938570653, 25.80311087558550395772637963059, 26.34864361989737304108618410229, 27.76661461943833732368562970127, 28.976781807886901143749875438995, 30.34309945286364937318197770363, 31.08897295299283581483819245761, 32.28824938091907360330123263713

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