↑

Properties

Label 1-1021-1021.256-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.794 - 0.607i$
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  + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯
L(s)  = 1  + (−0.201 − 0.979i)2-s + (0.445 + 0.895i)3-s + (−0.918 + 0.395i)4-s + (0.786 − 0.617i)5-s + (0.786 − 0.617i)6-s + (−0.542 − 0.840i)7-s + (0.572 + 0.819i)8-s + (−0.602 + 0.798i)9-s + (−0.763 − 0.645i)10-s + (0.237 − 0.971i)11-s + (−0.763 − 0.645i)12-s + (0.739 − 0.673i)13-s + (−0.713 + 0.700i)14-s + (0.903 + 0.429i)15-s + (0.687 − 0.726i)16-s + (−0.999 + 0.0369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.794 - 0.607i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.794 - 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3719668510 - 1.098537905i\)
\(L(\frac12)\) \(\approx\) \(0.3719668510 - 1.098537905i\)
\(L(1)\) \(\approx\) \(0.8598185285 - 0.5076109848i\)
\(L(1)\) \(\approx\) \(0.8598185285 - 0.5076109848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.201 - 0.979i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (0.786 - 0.617i)T \)
7 \( 1 + (-0.542 - 0.840i)T \)
11 \( 1 + (0.237 - 0.971i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (-0.999 + 0.0369i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (0.687 + 0.726i)T \)
29 \( 1 + (-0.542 + 0.840i)T \)
31 \( 1 + (0.0184 - 0.999i)T \)
37 \( 1 + (0.989 - 0.147i)T \)
41 \( 1 + (-0.602 - 0.798i)T \)
43 \( 1 + (-0.966 + 0.255i)T \)
47 \( 1 + (-0.713 + 0.700i)T \)
53 \( 1 + (-0.128 - 0.991i)T \)
59 \( 1 + (-0.886 - 0.462i)T \)
61 \( 1 + (-0.713 + 0.700i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.572 + 0.819i)T \)
73 \( 1 + (-0.713 - 0.700i)T \)
79 \( 1 + (0.997 + 0.0738i)T \)
83 \( 1 + (0.445 + 0.895i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (0.932 + 0.361i)T \)
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

−22.096292716050682468839080493945, −21.28848528997958504368367494094, −20.14395749039941570732210878982, −19.20925609271883935796161352526, −18.42246793925000794293508477231, −18.28609542523563133597185956892, −17.29099754997310768933127145508, −16.583608314151166645235727894784, −15.3010275612830065886667952190, −14.92151793606251062994214986955, −14.117669534575676620642007894778, −13.27096420750340367881762338071, −12.80998551918663545141057434582, −11.71367712701022295640967483455, −10.4429543552064252630858383765, −9.440345968513240704911354263746, −8.97016027919355095611378970211, −8.1141668694327987737842289938, −6.95746799891109467800539579375, −6.47327499247098464664897765013, −6.00776316259991145825908308699, −4.73595636136450941358635923781, −3.478829971468620786627290258762, −2.280578274415811227971336015861, −1.53575427541209636456811164508, 0.4873200144803015555327339599, 1.715124898050008163686641713028, 2.896966988142301392735554811706, 3.58870130489257678466737894062, 4.45922364305554849889682460865, 5.30345939263302321837796293440, 6.36218026673053790344310452673, 7.89617506358243684759714442490, 8.78554064839880225600276539098, 9.240759030010872393213202403351, 10.02341153815866505587843622220, 10.93674906193409929861166543381, 11.202566323576278162384263492968, 12.8578435899695746881341675781, 13.39296582617588240837980172840, 13.73868101874464821842219062303, 14.89113825319418897016047433149, 15.983377934582093864513885550339, 16.78700502378272455648347284738, 17.25039710353134335533244443961, 18.2191421578572939538235539918, 19.34121863993123769253006210994, 19.91702419570948417814495924034, 20.49384777415969500968132241774, 21.186164678331772402778552412325

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