↑

Properties

Label 1-42e2-1764.475-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.997 - 0.0676i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (−0.955 − 0.294i)5-s + (−0.0747 − 0.997i)11-s + (−0.826 + 0.563i)13-s + (−0.623 + 0.781i)17-s + 19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (0.623 − 0.781i)37-s + (−0.955 − 0.294i)41-s + (−0.955 + 0.294i)43-s + (0.0747 + 0.997i)47-s + (0.623 + 0.781i)53-s + (−0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)5-s + (−0.0747 − 0.997i)11-s + (−0.826 + 0.563i)13-s + (−0.623 + 0.781i)17-s + 19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (0.623 − 0.781i)37-s + (−0.955 − 0.294i)41-s + (−0.955 + 0.294i)43-s + (0.0747 + 0.997i)47-s + (0.623 + 0.781i)53-s + (−0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.997 - 0.0676i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.997 - 0.0676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050481359 - 0.03555977833i\)
\(L(\frac12)\) \(\approx\) \(1.050481359 - 0.03555977833i\)
\(L(1)\) \(\approx\) \(0.8524324584 - 0.03879038003i\)
\(L(1)\) \(\approx\) \(0.8524324584 - 0.03879038003i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.955 - 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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

−20.20196119708734589154195216331, −19.70755364753059244395117152417, −18.61520799358372161825750015979, −18.26010694685583665951562030005, −17.27062473462155622193762798617, −16.57138483048867426401894522138, −15.6507579190229700165884715143, −15.043911540261325181389932226248, −14.646153937480109322944212295161, −13.38192241709692403819049262017, −12.84371151453210839409286165887, −11.67775344548317837161002653347, −11.61746270508173458107571083716, −10.40796853178934327095487301822, −9.728953184233914175879862495755, −8.894680808239329627191462102671, −7.82809140807256685640090484655, −7.289020376893279417038430955063, −6.731445456516893922750409214358, −5.276665616958913811391231025643, −4.811887922358351111661442212682, −3.763811871257797798915086636729, −2.955237161294068238081520470490, −2.05300008000351836647060462070, −0.6197814386338392775168273682, 0.66556781018255759947518013569, 1.84729479188118683091348847497, 3.086402671469082061657253470230, 3.72599876578519111557328393839, 4.705671731979596898198037746639, 5.39346229577867714389908258915, 6.5033376678535607289030412797, 7.31181463495721333067945108118, 8.016340821444718717346044341415, 8.873523951987408986355254878471, 9.43423821011352805666212062169, 10.708102710129731876220961291594, 11.21265149733532453769592562955, 11.97317318316786248149370078938, 12.71603666608216478418165879164, 13.480550433766115263696259755138, 14.39647761619630102240014070769, 15.09488074450647087285843401596, 15.82519290824692319314530490242, 16.60555915429739119233671043806, 17.02757739215950698150982708007, 18.17464489994301819382708008982, 18.88585181577436116697437379387, 19.53265930695378869624443880002, 20.05548015649309185196990995293

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