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Properties

Label 1-1021-1021.8-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.997 + 0.0686i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.617 + 0.786i)2-s + (−0.932 + 0.361i)3-s + (−0.237 + 0.971i)4-s + (0.510 − 0.859i)5-s + (−0.859 − 0.510i)6-s + (0.999 − 0.0184i)7-s + (−0.911 + 0.412i)8-s + (0.739 − 0.673i)9-s + (0.991 − 0.128i)10-s + (−0.478 − 0.878i)11-s + (−0.128 − 0.991i)12-s + (0.961 − 0.273i)13-s + (0.631 + 0.775i)14-s + (−0.165 + 0.986i)15-s + (−0.886 − 0.462i)16-s + (0.918 − 0.395i)17-s + ⋯
L(s)  = 1  + (0.617 + 0.786i)2-s + (−0.932 + 0.361i)3-s + (−0.237 + 0.971i)4-s + (0.510 − 0.859i)5-s + (−0.859 − 0.510i)6-s + (0.999 − 0.0184i)7-s + (−0.911 + 0.412i)8-s + (0.739 − 0.673i)9-s + (0.991 − 0.128i)10-s + (−0.478 − 0.878i)11-s + (−0.128 − 0.991i)12-s + (0.961 − 0.273i)13-s + (0.631 + 0.775i)14-s + (−0.165 + 0.986i)15-s + (−0.886 − 0.462i)16-s + (0.918 − 0.395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.997 + 0.0686i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.997 + 0.0686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.464720172 + 0.08473171515i\)
\(L(\frac12)\) \(\approx\) \(2.464720172 + 0.08473171515i\)
\(L(1)\) \(\approx\) \(1.255054589 + 0.3968483424i\)
\(L(1)\) \(\approx\) \(1.255054589 + 0.3968483424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.617 + 0.786i)T \)
3 \( 1 + (-0.932 + 0.361i)T \)
5 \( 1 + (0.510 - 0.859i)T \)
7 \( 1 + (0.999 - 0.0184i)T \)
11 \( 1 + (-0.478 - 0.878i)T \)
13 \( 1 + (0.961 - 0.273i)T \)
17 \( 1 + (0.918 - 0.395i)T \)
19 \( 1 + (-0.673 + 0.739i)T \)
23 \( 1 + (-0.886 + 0.462i)T \)
29 \( 1 + (-0.0184 + 0.999i)T \)
31 \( 1 + (0.979 + 0.201i)T \)
37 \( 1 + (0.998 - 0.0554i)T \)
41 \( 1 + (-0.739 - 0.673i)T \)
43 \( 1 + (0.291 - 0.956i)T \)
47 \( 1 + (0.631 + 0.775i)T \)
53 \( 1 + (-0.147 + 0.989i)T \)
59 \( 1 + (-0.840 - 0.542i)T \)
61 \( 1 + (-0.631 - 0.775i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.412 - 0.911i)T \)
73 \( 1 + (0.631 - 0.775i)T \)
79 \( 1 + (-0.687 - 0.726i)T \)
83 \( 1 + (-0.932 + 0.361i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.798 - 0.602i)T \)
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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

−21.27563265167839983035911714523, −21.10072029080952660639696493945, −19.90302868665231894281600988103, −18.8145870626748288181000580142, −18.35841786172421416562803982296, −17.77524995443710624193595396944, −16.98957882195634785552077627378, −15.62820661436967259420094813525, −14.96498174379175317843556621208, −14.11059601633757580600815323689, −13.367482056054327048168252350918, −12.63905335507606805359931199053, −11.56088895181361776361168714806, −11.296705865660494157822585323358, −10.24425275036292070705362359913, −9.98241384155985849280251707882, −8.39141841234396203500730105577, −7.33242295890975637067319434665, −6.27712003828379701001112398392, −5.81459375449706696882460438642, −4.74189814867357524213619039239, −4.09570332335860732407558816821, −2.59184359963186475363383379108, −1.8960454532115154319357116095, −0.99683356836660511868235255462, 0.53759783867399876052166713673, 1.636849862358130030173303683075, 3.3195262921279609286653284188, 4.295302936281996072599287947380, 5.04515812423833471171652181797, 5.790250741313237816551925640616, 6.140632277487915277373885683634, 7.61435098329151359227752410218, 8.30093181457625308897870670095, 9.10195876166913818353709539517, 10.31198953114982475413026921051, 11.13262865221082623013592871754, 12.094000617414984782118094627817, 12.59296069527502697879809819166, 13.71280668296931288701840946352, 14.13190104447709567631997700305, 15.391286905993535673071018637392, 15.95037825197710551547859488827, 16.74860192285214483180103949951, 17.151370923463527732295824975347, 18.11826302836328044304015416067, 18.5445996266248690624395280139, 20.426528305965056671130589512204, 20.96895470594830296244848301134, 21.45246701095096688160065047214

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