↑

Properties

Label 1-1021-1021.176-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.162 + 0.986i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.869 + 0.494i)2-s + (0.445 − 0.895i)3-s + (0.510 + 0.859i)4-s + (0.0677 + 0.997i)5-s + (0.830 − 0.557i)6-s + (0.631 + 0.775i)7-s + (0.0184 + 0.999i)8-s + (−0.602 − 0.798i)9-s + (−0.434 + 0.900i)10-s + (0.612 + 0.790i)11-s + (0.997 − 0.0738i)12-s + (0.739 + 0.673i)13-s + (0.165 + 0.986i)14-s + (0.923 + 0.384i)15-s + (−0.478 + 0.878i)16-s + (0.141 − 0.989i)17-s + ⋯
L(s)  = 1  + (0.869 + 0.494i)2-s + (0.445 − 0.895i)3-s + (0.510 + 0.859i)4-s + (0.0677 + 0.997i)5-s + (0.830 − 0.557i)6-s + (0.631 + 0.775i)7-s + (0.0184 + 0.999i)8-s + (−0.602 − 0.798i)9-s + (−0.434 + 0.900i)10-s + (0.612 + 0.790i)11-s + (0.997 − 0.0738i)12-s + (0.739 + 0.673i)13-s + (0.165 + 0.986i)14-s + (0.923 + 0.384i)15-s + (−0.478 + 0.878i)16-s + (0.141 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.162 + 0.986i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.485860725 + 2.110055080i\)
\(L(\frac12)\) \(\approx\) \(2.485860725 + 2.110055080i\)
\(L(1)\) \(\approx\) \(1.975430723 + 0.8184052846i\)
\(L(1)\) \(\approx\) \(1.975430723 + 0.8184052846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.869 + 0.494i)T \)
3 \( 1 + (0.445 - 0.895i)T \)
5 \( 1 + (0.0677 + 0.997i)T \)
7 \( 1 + (0.631 + 0.775i)T \)
11 \( 1 + (0.612 + 0.790i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
17 \( 1 + (0.141 - 0.989i)T \)
19 \( 1 + (-0.389 - 0.920i)T \)
23 \( 1 + (-0.521 + 0.853i)T \)
29 \( 1 + (-0.987 - 0.159i)T \)
31 \( 1 + (0.755 + 0.655i)T \)
37 \( 1 + (0.0431 - 0.999i)T \)
41 \( 1 + (-0.389 - 0.920i)T \)
43 \( 1 + (0.892 + 0.451i)T \)
47 \( 1 + (0.771 - 0.636i)T \)
53 \( 1 + (-0.972 - 0.231i)T \)
59 \( 1 + (0.963 - 0.267i)T \)
61 \( 1 + (-0.936 - 0.349i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.0184 + 0.999i)T \)
73 \( 1 + (0.165 - 0.986i)T \)
79 \( 1 + (0.237 - 0.971i)T \)
83 \( 1 + (-0.998 + 0.0615i)T \)
89 \( 1 + (0.816 - 0.577i)T \)
97 \( 1 + (-0.153 + 0.988i)T \)
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

−21.13364357533247003175063108400, −20.80482134018735921658700140717, −20.20315074410881509707239241760, −19.53533580079214545938943606401, −18.5959202249416646468186643, −17.0549911071438461500949553121, −16.70628035213090916114286107890, −15.79440675834371189825216552262, −14.97275514188354631129674218126, −14.22322539428842477288343300724, −13.59541648433730238932394185186, −12.84678037724656290629070711063, −11.84519802297217444036370897143, −10.9386587675950032712455339934, −10.4121191381545731724117401761, −9.51787538330981012788127486376, −8.436390260629093166486046860989, −7.940744928633990118250025228957, −6.181186259192342750635880678968, −5.618758942833334162932306539302, −4.491879463196633796778673551226, −4.05045251694627471179730729518, −3.299595141857867069421503196850, −1.89839135433749416266075862461, −0.99955711833974936225774928200, 1.79113265283807515104932525685, 2.35548689813828047773370704286, 3.327841872154740839875292779114, 4.2842414916839414204862861683, 5.51943897141589223380031719455, 6.30334183170231178863791822126, 7.0891734898965968206972452639, 7.564530104476129484017107926697, 8.681607310630346810350153248506, 9.39749834268520616057887622187, 11.06885089521124284997644339649, 11.64933309580010991520778203723, 12.23405261729997817015919706202, 13.30112361508523932288338532941, 14.03907951781082267455207886886, 14.48581571119626373580465735783, 15.26760920080466478978426789960, 15.90406848674117606901440987517, 17.426620717658367699013659167472, 17.726859950509830333094589016966, 18.605566336007819113403737276840, 19.37515336108904249922156482002, 20.36650501005652580139948116937, 21.11410391118556108161311592993, 21.90468884400091476539292557130

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