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Properties

Label 1-1080-1080.779-r0-0-0
Degree $1$
Conductor $1080$
Sign $0.0581 - 0.998i$
Analytic cond. $5.01549$
Root an. cond. $5.01549$
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  + (0.173 − 0.984i)7-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 − 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s + (−0.939 − 0.342i)49-s − 53-s + (0.939 + 0.342i)59-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)7-s + (0.939 − 0.342i)11-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)23-s + (0.766 + 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 − 0.866i)37-s + (−0.766 + 0.642i)41-s + (0.939 − 0.342i)43-s + (−0.173 + 0.984i)47-s + (−0.939 − 0.342i)49-s − 53-s + (0.939 + 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.0581 - 0.998i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ 0.0581 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076066044 - 1.015215953i\)
\(L(\frac12)\) \(\approx\) \(1.076066044 - 1.015215953i\)
\(L(1)\) \(\approx\) \(1.078561886 - 0.3107206436i\)
\(L(1)\) \(\approx\) \(1.078561886 - 0.3107206436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−21.68774224645011342616057908995, −21.05034987209719964071023739288, −19.95585871707864397745680038750, −19.310695292960568057267541571432, −18.64397908758605130254313366365, −17.56183022231604609015456676031, −17.2749469046783985565856166952, −15.96435730902997519285926407125, −15.46746905973855194916019281829, −14.65387293795771125032469520942, −13.81734805443722565401746696782, −12.95818397762381805065433363944, −12.0178196038638314927820231471, −11.49292669999381032463300100984, −10.58113036665753825902507895913, −9.44945438005482870730483413644, −8.829616694839150560623727394924, −8.17301508046305826878076411589, −6.78414064486661626512798356623, −6.33636272834122171054653233817, −5.25382397561035618669505684726, −4.31278309853223916834636932141, −3.416048541888128396150163415436, −2.17288330910317246769246328197, −1.428107569398971784550107136791, 0.64551530689703670042407440860, 1.65264992312214082800654685000, 3.001961136294995283371900747273, 3.93625511204138573504670485473, 4.61200633253970991381231393994, 5.89029059635780551253813378270, 6.5991108951809845350844234203, 7.51801308404325727715725978519, 8.397494953261129609019986244730, 9.17858198169867109352752868127, 10.27538001329205608901729452542, 10.846675206195139905639216753609, 11.68669027374078114947089544630, 12.63868578650222672895595243068, 13.49516773059655179116804850417, 14.19212179486218810173781036707, 14.83425359179480194793454515071, 16.05009780486043841858322323220, 16.504216010430307837396770105881, 17.41815052875239580164461295893, 18.07088074382353564186182110949, 19.001382231100772327475640567674, 19.80316097416096377479935003192, 20.56018179411047618686621064209, 21.00470070541106856468971511975

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