↑

Properties

Label 1-2028-2028.1043-r0-0-0
Degree $1$
Conductor $2028$
Sign $0.557 - 0.829i$
Analytic cond. $9.41799$
Root an. cond. $9.41799$
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  + (0.354 − 0.935i)5-s + (0.919 − 0.391i)7-s + (−0.845 + 0.534i)11-s + (0.919 − 0.391i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.748 − 0.663i)25-s + (0.0402 − 0.999i)29-s + (0.748 − 0.663i)31-s + (−0.0402 − 0.999i)35-s + (−0.200 − 0.979i)37-s + (−0.278 + 0.960i)41-s + (0.200 − 0.979i)43-s + (0.568 + 0.822i)47-s + (0.692 − 0.721i)49-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)5-s + (0.919 − 0.391i)7-s + (−0.845 + 0.534i)11-s + (0.919 − 0.391i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.748 − 0.663i)25-s + (0.0402 − 0.999i)29-s + (0.748 − 0.663i)31-s + (−0.0402 − 0.999i)35-s + (−0.200 − 0.979i)37-s + (−0.278 + 0.960i)41-s + (0.200 − 0.979i)43-s + (0.568 + 0.822i)47-s + (0.692 − 0.721i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.557 - 0.829i$
Analytic conductor: \(9.41799\)
Root analytic conductor: \(9.41799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (0:\ ),\ 0.557 - 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.707659501 - 0.9097891814i\)
\(L(\frac12)\) \(\approx\) \(1.707659501 - 0.9097891814i\)
\(L(1)\) \(\approx\) \(1.231654576 - 0.2772769401i\)
\(L(1)\) \(\approx\) \(1.231654576 - 0.2772769401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.354 - 0.935i)T \)
7 \( 1 + (0.919 - 0.391i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
17 \( 1 + (0.919 - 0.391i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.0402 - 0.999i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (-0.200 - 0.979i)T \)
41 \( 1 + (-0.278 + 0.960i)T \)
43 \( 1 + (0.200 - 0.979i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (-0.120 + 0.992i)T \)
59 \( 1 + (0.987 - 0.160i)T \)
61 \( 1 + (0.799 - 0.600i)T \)
67 \( 1 + (0.996 - 0.0804i)T \)
71 \( 1 + (0.692 + 0.721i)T \)
73 \( 1 + (0.885 + 0.464i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (-0.970 - 0.239i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.632 + 0.774i)T \)
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

−20.05468297007683551449635811682, −19.08381086273990351738452931303, −18.50435529666834164299269275986, −17.98138619240162376966643708411, −17.33156537637523987101054101483, −16.34839123704520512680592558878, −15.55974063999776909773543199728, −14.86888390067786258680293479240, −14.18356071031593352661517777056, −13.66955056046056547522298488428, −12.659544087867101494899131714549, −11.81415218446975157093097092873, −11.08471304537993545814640807019, −10.457466906336442365387180658813, −9.81555230947675519175140248826, −8.62297656435348165427954619847, −8.12439542642552480690674958055, −7.20433196723483649760170966183, −6.44486294567251649464491050147, −5.45013351195784687061050672957, −5.017471902476971383190988231538, −3.71517462142439871193090089498, −2.84625622989238236126072905325, −2.200284425806540832324488967073, −1.0609757991909926121393302060, 0.770729392350026782574299914488, 1.65700444319714260092884950213, 2.47379220946308830490227627787, 3.80484540193180753082531828231, 4.5293895725496073222890418178, 5.40105540789910838305072531462, 5.78828625795255702206161929404, 7.22877371378062963466668621197, 7.91911638963123212781061655298, 8.33048760101393811226597543247, 9.679119186296266573388083446, 9.86166745421379000526661727393, 10.93418152422547241592200608108, 11.85045035275047371357072432682, 12.35616588534223311024026757094, 13.30038220937944595438378709907, 13.895296942491519954799592876312, 14.5656053042699433130733422253, 15.61555123593950386685103283502, 16.10750232795257889410394763653, 17.15428328372912389885400524539, 17.41316018866725210233635972884, 18.31221787948735350233111670144, 18.97745812075311369164783160889, 20.14726177953175947147099818287

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