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Properties

Label 1-1021-1021.115-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.695 + 0.718i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0184 − 0.999i)2-s + (−0.982 − 0.183i)3-s + (−0.999 − 0.0369i)4-s + (−0.201 + 0.979i)5-s + (−0.201 + 0.979i)6-s + (−0.886 − 0.462i)7-s + (−0.0554 + 0.998i)8-s + (0.932 + 0.361i)9-s + (0.975 + 0.219i)10-s + (−0.918 − 0.395i)11-s + (0.975 + 0.219i)12-s + (−0.602 − 0.798i)13-s + (−0.478 + 0.878i)14-s + (0.378 − 0.925i)15-s + (0.997 + 0.0738i)16-s + (−0.412 − 0.911i)17-s + ⋯
L(s)  = 1  + (0.0184 − 0.999i)2-s + (−0.982 − 0.183i)3-s + (−0.999 − 0.0369i)4-s + (−0.201 + 0.979i)5-s + (−0.201 + 0.979i)6-s + (−0.886 − 0.462i)7-s + (−0.0554 + 0.998i)8-s + (0.932 + 0.361i)9-s + (0.975 + 0.219i)10-s + (−0.918 − 0.395i)11-s + (0.975 + 0.219i)12-s + (−0.602 − 0.798i)13-s + (−0.478 + 0.878i)14-s + (0.378 − 0.925i)15-s + (0.997 + 0.0738i)16-s + (−0.412 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1446175215 + 0.06131864469i\)
\(L(\frac12)\) \(\approx\) \(0.1446175215 + 0.06131864469i\)
\(L(1)\) \(\approx\) \(0.4291228210 - 0.2375844480i\)
\(L(1)\) \(\approx\) \(0.4291228210 - 0.2375844480i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.0184 - 0.999i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.201 + 0.979i)T \)
7 \( 1 + (-0.886 - 0.462i)T \)
11 \( 1 + (-0.918 - 0.395i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (-0.412 - 0.911i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.997 - 0.0738i)T \)
29 \( 1 + (-0.886 + 0.462i)T \)
31 \( 1 + (-0.542 - 0.840i)T \)
37 \( 1 + (-0.128 - 0.991i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (0.165 - 0.986i)T \)
47 \( 1 + (-0.478 + 0.878i)T \)
53 \( 1 + (-0.763 + 0.645i)T \)
59 \( 1 + (0.687 + 0.726i)T \)
61 \( 1 + (-0.478 + 0.878i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.0554 + 0.998i)T \)
73 \( 1 + (-0.478 - 0.878i)T \)
79 \( 1 + (-0.659 - 0.751i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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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

−21.63872089104119182078753765004, −21.08604157487255654731787912148, −19.787566081968367899393184746, −18.90158409344708170652989258963, −18.21578935960284150585260800002, −17.256973076101945360860448510180, −16.75189021881541970398961169353, −16.02108164624411355872897418747, −15.58214165912488239877838050568, −14.72302705260029300889193372543, −13.32128100479105287964879320672, −12.80142039131559135514719309850, −12.27097254368611909005989051942, −11.18336387990527450053828881774, −9.80961594806764770690323932039, −9.55400803786077633749853933920, −8.499667646658573936453659813398, −7.48817331711321413603698217760, −6.69869718881759871921385292063, −5.79453567664749391336270471986, −5.11439951724307874806955934159, −4.47984630983403558638275303829, −3.41034483779963122798425215803, −1.579581118533379479642400778205, −0.11034134171449974655419546836, 0.80410943260546728127301601914, 2.4729992761194768190549536868, 3.06777477565519139150127263822, 4.0972783962202529996952067378, 5.2392754582026518886095670365, 5.86045176826542407505164742348, 7.28053847163467882712048770369, 7.48493544251229209649946576052, 9.208157326513996802824753226058, 9.96401882825773681781675662567, 10.75680470593515055265916102994, 11.09499779870046381787589151244, 12.03551023011762478269619223417, 12.99161506605921015735867826925, 13.34863319148620234855712166558, 14.41874439380699282242985155932, 15.536261570816727407232505045262, 16.24692492219310969292559999954, 17.33033728399599193812033532137, 17.97443980524797929805769862149, 18.66435004206589327295226220974, 19.199217546430871143853604345705, 20.10813776163767758821536643753, 20.93431423986450941015839210693, 22.03784713360971440068883809966

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