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Properties

Label 1-1021-1021.15-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.859 + 0.510i$
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  + (0.0554 − 0.998i)2-s + (−0.850 − 0.526i)3-s + (−0.993 − 0.110i)4-s + (0.423 + 0.905i)5-s + (−0.572 + 0.819i)6-s + (0.128 + 0.991i)7-s + (−0.165 + 0.986i)8-s + (0.445 + 0.895i)9-s + (0.927 − 0.372i)10-s + (0.985 + 0.171i)11-s + (0.786 + 0.617i)12-s + (−0.932 + 0.361i)13-s + (0.997 − 0.0738i)14-s + (0.116 − 0.993i)15-s + (0.975 + 0.219i)16-s + (−0.225 − 0.974i)17-s + ⋯
L(s)  = 1  + (0.0554 − 0.998i)2-s + (−0.850 − 0.526i)3-s + (−0.993 − 0.110i)4-s + (0.423 + 0.905i)5-s + (−0.572 + 0.819i)6-s + (0.128 + 0.991i)7-s + (−0.165 + 0.986i)8-s + (0.445 + 0.895i)9-s + (0.927 − 0.372i)10-s + (0.985 + 0.171i)11-s + (0.786 + 0.617i)12-s + (−0.932 + 0.361i)13-s + (0.997 − 0.0738i)14-s + (0.116 − 0.993i)15-s + (0.975 + 0.219i)16-s + (−0.225 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.859 + 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8213421970 + 0.2256380242i\)
\(L(\frac12)\) \(\approx\) \(0.8213421970 + 0.2256380242i\)
\(L(1)\) \(\approx\) \(0.7643185971 - 0.1851090965i\)
\(L(1)\) \(\approx\) \(0.7643185971 - 0.1851090965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.0554 - 0.998i)T \)
3 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (0.423 + 0.905i)T \)
7 \( 1 + (0.128 + 0.991i)T \)
11 \( 1 + (0.985 + 0.171i)T \)
13 \( 1 + (-0.932 + 0.361i)T \)
17 \( 1 + (-0.225 - 0.974i)T \)
19 \( 1 + (-0.552 - 0.833i)T \)
23 \( 1 + (-0.678 - 0.734i)T \)
29 \( 1 + (0.923 - 0.384i)T \)
31 \( 1 + (0.622 + 0.782i)T \)
37 \( 1 + (0.990 + 0.135i)T \)
41 \( 1 + (0.552 + 0.833i)T \)
43 \( 1 + (0.521 + 0.853i)T \)
47 \( 1 + (-0.562 - 0.826i)T \)
53 \( 1 + (-0.489 + 0.872i)T \)
59 \( 1 + (0.177 + 0.984i)T \)
61 \( 1 + (-0.434 + 0.900i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.165 - 0.986i)T \)
73 \( 1 + (0.997 + 0.0738i)T \)
79 \( 1 + (0.830 - 0.557i)T \)
83 \( 1 + (-0.0307 + 0.999i)T \)
89 \( 1 + (-0.952 - 0.303i)T \)
97 \( 1 + (-0.650 + 0.759i)T \)
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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

−21.74136285355691363773265346515, −21.01884372844802545859805535456, −19.98586903350082319666115005139, −19.19164047128808929440701309361, −17.747695192535271370618770492452, −17.36154471353627713860209622993, −16.89403171867370941389274220106, −16.2902508666645244680094350650, −15.40633109937626249372607981014, −14.480774525021064571962830350305, −13.840031193652599899524203841857, −12.6919407723635412831433475802, −12.31794451863137282339515343952, −11.0400766321847945860522763862, −9.9270818769162211514032008050, −9.645241029912833496112490088930, −8.48375388488527784393636448617, −7.6803623785761535883845218940, −6.52112952517248401317967714391, −6.00179724651124463043831254561, −5.05742975512042876179642975638, −4.2660417930342128319297796680, −3.77675156989063141127618290046, −1.52169229915306517834078439300, −0.45958855089083128429641152562, 1.18345766843810110863797914502, 2.42355925626561283534144511276, 2.6255177352807461067022018056, 4.36144102683541161239641162006, 4.997534020190882989251234682254, 6.15156090645072499374304484717, 6.68416800097506011901160762651, 7.87203909242668642125011339445, 9.07415052843085769104133253949, 9.76437233046071644743212087656, 10.622623800464742758507922826604, 11.558793283900132128182250327523, 11.86495848449925335297455081292, 12.66283426854644466962214260808, 13.66026561603986176044631383641, 14.363234084963085629232902478516, 15.12454003422299544240847178646, 16.41675711791883045334187288290, 17.430754795701445060796968457419, 17.94666285094912905306499430146, 18.4641961009862438509708994753, 19.392119463520560607513332579037, 19.74306241367862373706414727602, 21.34508772707103662219575012251, 21.6413454861761645107181980786

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