↑

Properties

Label 1-2028-2028.1055-r1-0-0
Degree $1$
Conductor $2028$
Sign $0.969 - 0.245i$
Analytic cond. $217.938$
Root an. cond. $217.938$
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  + (−0.822 + 0.568i)5-s + (0.316 − 0.948i)7-s + (0.721 + 0.692i)11-s + (0.948 + 0.316i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)23-s + (0.354 − 0.935i)25-s + (−0.278 + 0.960i)29-s + (0.935 − 0.354i)31-s + (0.278 + 0.960i)35-s + (−0.160 + 0.987i)37-s + (0.391 − 0.919i)41-s + (0.987 − 0.160i)43-s + (−0.464 + 0.885i)47-s + (−0.799 − 0.600i)49-s + ⋯
L(s)  = 1  + (−0.822 + 0.568i)5-s + (0.316 − 0.948i)7-s + (0.721 + 0.692i)11-s + (0.948 + 0.316i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)23-s + (0.354 − 0.935i)25-s + (−0.278 + 0.960i)29-s + (0.935 − 0.354i)31-s + (0.278 + 0.960i)35-s + (−0.160 + 0.987i)37-s + (0.391 − 0.919i)41-s + (0.987 − 0.160i)43-s + (−0.464 + 0.885i)47-s + (−0.799 − 0.600i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(217.938\)
Root analytic conductor: \(217.938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (1:\ ),\ 0.969 - 0.245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.350518860 - 0.2927972521i\)
\(L(\frac12)\) \(\approx\) \(2.350518860 - 0.2927972521i\)
\(L(1)\) \(\approx\) \(1.137702414 + 0.01113973002i\)
\(L(1)\) \(\approx\) \(1.137702414 + 0.01113973002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (0.316 - 0.948i)T \)
11 \( 1 + (0.721 + 0.692i)T \)
17 \( 1 + (0.948 + 0.316i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.278 + 0.960i)T \)
31 \( 1 + (0.935 - 0.354i)T \)
37 \( 1 + (-0.160 + 0.987i)T \)
41 \( 1 + (0.391 - 0.919i)T \)
43 \( 1 + (0.987 - 0.160i)T \)
47 \( 1 + (-0.464 + 0.885i)T \)
53 \( 1 + (0.748 - 0.663i)T \)
59 \( 1 + (0.903 + 0.428i)T \)
61 \( 1 + (-0.200 + 0.979i)T \)
67 \( 1 + (-0.534 - 0.845i)T \)
71 \( 1 + (-0.600 - 0.799i)T \)
73 \( 1 + (0.239 - 0.970i)T \)
79 \( 1 + (-0.885 - 0.464i)T \)
83 \( 1 + (-0.992 - 0.120i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.0804 + 0.996i)T \)
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

−19.59542071188620198456039859618, −19.11757069989174715420391945944, −18.487590922056164125713840592057, −17.54263649928575281669835523038, −16.78288445460678057400694557756, −16.03655806887171093973194606869, −15.55495319367847282003227309012, −14.64834050246291884433889728694, −14.04828527905337102217954995055, −13.042024532232055668394911850488, −12.23770244557187856772805206800, −11.613920506851188680943663420041, −11.30439248005700903837825767051, −9.9245685532076021133899814137, −9.238580617681664105008400469939, −8.501088670150381779660140606455, −7.87107139447143048625904862969, −7.08622842719048583592193570730, −5.8145728663468338981760754953, −5.435535267765995644121139582334, −4.388785056566515587813036988523, −3.54859543647354310952268535753, −2.78135924198235929023562740037, −1.45207990254035942872182885827, −0.75024670012668082555450946285, 0.6258810251111736883819383622, 1.39597618622460456836598799094, 2.73199315640409123743612855658, 3.55130036556537906010318288220, 4.28900782778843625781126103041, 4.983506625750212956952363444033, 6.24705625344823005511623160069, 7.1000754056506943268960236702, 7.47190538604952257864128709661, 8.339353576583736057300759255522, 9.30048876667910443171872644765, 10.25936707067748273202410641325, 10.73153963638368626041014460106, 11.677372424423868206300522527229, 12.14086907716955573454341598888, 13.11032293863377072017020141867, 14.06935225930156408435592909458, 14.589037383192043921126142978162, 15.19170551807341858215615754800, 16.15262764475271109202819794898, 16.77041187985226409533705640538, 17.56440132907786492264710708989, 18.217801174750150190798627029686, 19.19623500657423258635326521161, 19.576198655916029129172192384570

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