Properties

Label 1-1021-1021.925-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.994 + 0.102i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (0.183 + 0.982i)2-s + (0.273 + 0.961i)3-s + (−0.932 + 0.361i)4-s + (0.445 + 0.895i)5-s + (−0.895 + 0.445i)6-s + (−0.995 + 0.0922i)7-s + (−0.526 − 0.850i)8-s + (−0.850 + 0.526i)9-s + (−0.798 + 0.602i)10-s + (−0.602 + 0.798i)11-s + (−0.602 − 0.798i)12-s + (−0.183 + 0.982i)13-s + (−0.273 − 0.961i)14-s + (−0.739 + 0.673i)15-s + (0.739 − 0.673i)16-s + (−0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.982i)2-s + (0.273 + 0.961i)3-s + (−0.932 + 0.361i)4-s + (0.445 + 0.895i)5-s + (−0.895 + 0.445i)6-s + (−0.995 + 0.0922i)7-s + (−0.526 − 0.850i)8-s + (−0.850 + 0.526i)9-s + (−0.798 + 0.602i)10-s + (−0.602 + 0.798i)11-s + (−0.602 − 0.798i)12-s + (−0.183 + 0.982i)13-s + (−0.273 − 0.961i)14-s + (−0.739 + 0.673i)15-s + (0.739 − 0.673i)16-s + (−0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (925, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.994 + 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7420366760 - 0.03822790650i\)
\(L(\frac12)\) \(\approx\) \(-0.7420366760 - 0.03822790650i\)
\(L(1)\) \(\approx\) \(0.2445292825 + 0.8163670495i\)
\(L(1)\) \(\approx\) \(0.2445292825 + 0.8163670495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.183 + 0.982i)T \)
3 \( 1 + (0.273 + 0.961i)T \)
5 \( 1 + (0.445 + 0.895i)T \)
7 \( 1 + (-0.995 + 0.0922i)T \)
11 \( 1 + (-0.602 + 0.798i)T \)
13 \( 1 + (-0.183 + 0.982i)T \)
17 \( 1 + (-0.445 - 0.895i)T \)
19 \( 1 + (-0.526 + 0.850i)T \)
23 \( 1 + (0.739 + 0.673i)T \)
29 \( 1 + (-0.0922 + 0.995i)T \)
31 \( 1 + (-0.526 - 0.850i)T \)
37 \( 1 + (-0.961 + 0.273i)T \)
41 \( 1 + (0.850 + 0.526i)T \)
43 \( 1 + (-0.995 + 0.0922i)T \)
47 \( 1 + (-0.273 - 0.961i)T \)
53 \( 1 + (0.673 - 0.739i)T \)
59 \( 1 + (-0.961 + 0.273i)T \)
61 \( 1 + (0.273 + 0.961i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.850 + 0.526i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (0.602 + 0.798i)T \)
83 \( 1 + (0.273 + 0.961i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (0.995 - 0.0922i)T \)
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

−20.47002891355413099127387650234, −19.767091828547043530159681855800, −19.27795930485006428801603415797, −18.53298809896830311349044739712, −17.515150212891952167406544030540, −17.13894570142039162198188338677, −15.874397522513633216707584728, −14.87735603293818980537270437802, −13.74052901547014297173949380455, −13.21704691756082358408522944612, −12.76736611841747824588815573769, −12.22286350904306169558300765655, −10.94960446019549407327079712481, −10.31345706812343241353342355431, −9.11617819545337323778128964438, −8.72118229455120821063601512685, −7.81115095518643941549892674669, −6.40146059618683874496159532720, −5.75945639198210968095153216253, −4.808965324142821957028880780931, −3.51018528929064856856435982621, −2.754756926942682352721427666973, −1.892450967767553389015333494970, −0.66272779553520012172574483314, −0.20217996821585554774836804134, 2.21710652927690406698596602224, 3.189075020611873331423432414410, 3.95483326349731870865279043061, 4.993120686276458696638248958484, 5.750725317232746935763630182197, 6.80004944791025352729455981687, 7.26817050054865093748116684737, 8.5471541842835886763165932244, 9.51918166037893979924109847427, 9.76824465912195960335572596141, 10.74244266011613970164761705138, 11.89614904656505492955073751058, 13.09994047427180691677796045782, 13.68382621609072207920481485444, 14.628372722539849087349673008808, 15.05332875879036896877063363622, 15.887290839284973074977821109020, 16.52674435825502957448645131518, 17.25017002608684824776714413869, 18.276948977982416114945891340541, 18.867591298885538473565980341074, 19.82793328036691601861262317599, 20.9865103604557878694517938625, 21.59388499812272909225748439020, 22.3234702697197672579260080731

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