↑

Properties

Label 1-1021-1021.69-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.662 + 0.748i$
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  + (−0.237 + 0.971i)2-s + (0.739 − 0.673i)3-s + (−0.886 − 0.462i)4-s + (−0.521 + 0.853i)5-s + (0.478 + 0.878i)6-s + (0.999 − 0.0369i)7-s + (0.659 − 0.751i)8-s + (0.0922 − 0.995i)9-s + (−0.704 − 0.709i)10-s + (0.998 + 0.0492i)11-s + (−0.966 + 0.255i)12-s + (0.850 − 0.526i)13-s + (−0.201 + 0.979i)14-s + (0.189 + 0.981i)15-s + (0.572 + 0.819i)16-s + (0.285 + 0.958i)17-s + ⋯
L(s)  = 1  + (−0.237 + 0.971i)2-s + (0.739 − 0.673i)3-s + (−0.886 − 0.462i)4-s + (−0.521 + 0.853i)5-s + (0.478 + 0.878i)6-s + (0.999 − 0.0369i)7-s + (0.659 − 0.751i)8-s + (0.0922 − 0.995i)9-s + (−0.704 − 0.709i)10-s + (0.998 + 0.0492i)11-s + (−0.966 + 0.255i)12-s + (0.850 − 0.526i)13-s + (−0.201 + 0.979i)14-s + (0.189 + 0.981i)15-s + (0.572 + 0.819i)16-s + (0.285 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.662 + 0.748i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.662 + 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.640840168 + 0.7389307230i\)
\(L(\frac12)\) \(\approx\) \(1.640840168 + 0.7389307230i\)
\(L(1)\) \(\approx\) \(1.191560549 + 0.3951592143i\)
\(L(1)\) \(\approx\) \(1.191560549 + 0.3951592143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.237 + 0.971i)T \)
3 \( 1 + (0.739 - 0.673i)T \)
5 \( 1 + (-0.521 + 0.853i)T \)
7 \( 1 + (0.999 - 0.0369i)T \)
11 \( 1 + (0.998 + 0.0492i)T \)
13 \( 1 + (0.850 - 0.526i)T \)
17 \( 1 + (0.285 + 0.958i)T \)
19 \( 1 + (-0.816 + 0.577i)T \)
23 \( 1 + (0.423 + 0.905i)T \)
29 \( 1 + (0.531 - 0.846i)T \)
31 \( 1 + (-0.801 + 0.597i)T \)
37 \( 1 + (-0.592 - 0.805i)T \)
41 \( 1 + (0.816 - 0.577i)T \)
43 \( 1 + (-0.0677 + 0.997i)T \)
47 \( 1 + (-0.747 - 0.664i)T \)
53 \( 1 + (0.730 - 0.682i)T \)
59 \( 1 + (0.582 - 0.812i)T \)
61 \( 1 + (0.949 - 0.314i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.659 + 0.751i)T \)
73 \( 1 + (-0.201 - 0.979i)T \)
79 \( 1 + (-0.0554 + 0.998i)T \)
83 \( 1 + (0.213 + 0.976i)T \)
89 \( 1 + (0.552 + 0.833i)T \)
97 \( 1 + (-0.969 + 0.243i)T \)
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

−21.25505212434217991731434769326, −20.52353407485998632293071379917, −20.27073984622590074823088419273, −19.3037640229044548382853111837, −18.71288362828056448552144032224, −17.64236520975208335555562093611, −16.72733935100319901233569480689, −16.227155434739239727812523611, −15.00877869935773256297937257241, −14.309097674903199077484883223533, −13.579377891521480557741245720887, −12.69374827896375856853282741518, −11.650060555098334472347270618151, −11.23412493597190807811830351727, −10.30544551950098936896078255805, −9.08286828867286115884128043048, −8.86707866699091544687623232860, −8.19708353846014296512661570016, −7.14182838855636183694878750925, −5.33582675834875317881716761309, −4.45475993011265272869921297051, −4.12230576340250148193585986431, −3.03382590580553161054227912490, −1.89534799023184614977139800928, −1.03142135011416389721207427448, 1.0830441470815831346589573803, 1.99927469078000656335399397524, 3.67181602161676587821815076120, 3.95619209278693760001029860646, 5.53044565551910003306412650922, 6.43992248602004205797504813615, 7.06231497158677547032314113420, 8.128182025806076941440257867720, 8.22978779154807445837685507726, 9.306188479378982857740413138, 10.41958035368135528400825977668, 11.25920719555654960452283580447, 12.31790870001017710967057250892, 13.26155171006129567427351065317, 14.19333456880860162513570273772, 14.65669113148210931185678448378, 15.123611434804979824731839424569, 16.0456325076029586427898433690, 17.33588424493396194200103546215, 17.73475442605551016552057743001, 18.517231477669089075733079153327, 19.33173061663711925913302922387, 19.67112953984242918232912743150, 20.96149928072902840017420483525, 21.72477750362671022709311348604

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