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Properties

Label 1-1021-1021.25-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.587 - 0.808i$
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  + (0.165 + 0.986i)2-s + (0.0922 + 0.995i)3-s + (−0.945 + 0.326i)4-s + (0.704 + 0.709i)5-s + (−0.966 + 0.255i)6-s + (0.378 − 0.925i)7-s + (−0.478 − 0.878i)8-s + (−0.982 + 0.183i)9-s + (−0.582 + 0.812i)10-s + (−0.862 − 0.505i)11-s + (−0.412 − 0.911i)12-s + (0.445 + 0.895i)13-s + (0.975 + 0.219i)14-s + (−0.641 + 0.767i)15-s + (0.786 − 0.617i)16-s + (0.355 + 0.934i)17-s + ⋯
L(s)  = 1  + (0.165 + 0.986i)2-s + (0.0922 + 0.995i)3-s + (−0.945 + 0.326i)4-s + (0.704 + 0.709i)5-s + (−0.966 + 0.255i)6-s + (0.378 − 0.925i)7-s + (−0.478 − 0.878i)8-s + (−0.982 + 0.183i)9-s + (−0.582 + 0.812i)10-s + (−0.862 − 0.505i)11-s + (−0.412 − 0.911i)12-s + (0.445 + 0.895i)13-s + (0.975 + 0.219i)14-s + (−0.641 + 0.767i)15-s + (0.786 − 0.617i)16-s + (0.355 + 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.587 - 0.808i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.587 - 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5153293255 + 1.011437093i\)
\(L(\frac12)\) \(\approx\) \(-0.5153293255 + 1.011437093i\)
\(L(1)\) \(\approx\) \(0.5301741971 + 0.8909868882i\)
\(L(1)\) \(\approx\) \(0.5301741971 + 0.8909868882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.165 + 0.986i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (0.704 + 0.709i)T \)
7 \( 1 + (0.378 - 0.925i)T \)
11 \( 1 + (-0.862 - 0.505i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
17 \( 1 + (0.355 + 0.934i)T \)
19 \( 1 + (0.332 + 0.943i)T \)
23 \( 1 + (-0.927 + 0.372i)T \)
29 \( 1 + (-0.990 - 0.135i)T \)
31 \( 1 + (-0.823 + 0.567i)T \)
37 \( 1 + (0.801 + 0.597i)T \)
41 \( 1 + (0.332 + 0.943i)T \)
43 \( 1 + (-0.434 - 0.900i)T \)
47 \( 1 + (-0.678 + 0.734i)T \)
53 \( 1 + (0.531 - 0.846i)T \)
59 \( 1 + (-0.999 - 0.0123i)T \)
61 \( 1 + (-0.297 - 0.954i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (-0.478 - 0.878i)T \)
73 \( 1 + (0.975 - 0.219i)T \)
79 \( 1 + (-0.201 + 0.979i)T \)
83 \( 1 + (-0.908 - 0.417i)T \)
89 \( 1 + (-0.389 - 0.920i)T \)
97 \( 1 + (0.881 + 0.473i)T \)
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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

−20.84813463977726648619848809027, −20.38276066132180311742968383686, −19.74812011992777423123024317541, −18.497650667631716398206980958145, −18.121059386361233319927640098296, −17.79496739036678131726958796032, −16.60028338661913795598138401291, −15.382900611155045526929362235758, −14.487819518031266507893274898220, −13.59517609177224067103334755404, −13.00955100938129765116716333739, −12.47988483377500487498019072258, −11.70383376578865960211871479077, −10.87395830988045330937286292215, −9.73579129077428908591953665632, −9.04590312811615472904018359290, −8.27757902832113168348754250025, −7.4470301562740858960538654704, −5.725543772007886805822220736499, −5.612248007198239362159982124, −4.5726100968476094508629268835, −2.98553875669091772676887737786, −2.35731084858171166905915412077, −1.595063378263288178320356291651, −0.432197814804568822802527975244, 1.71272244321555218935148338074, 3.33107933840244706230730208730, 3.81756624722923359579165092810, 4.86514026324975886308973321354, 5.77810469832729248061124758880, 6.334244135718601445430975299980, 7.59684832637815643088129829777, 8.16859542861871233742758432303, 9.271530116717525028485454400957, 10.0367688689100577544636358854, 10.64084245257105861619654559791, 11.53949669162974748128505749523, 13.04394946189979942316117967565, 13.79221009140845280332803026883, 14.34192069172480390737150312348, 14.90166291611286556746062106960, 15.909761358652830771909346125228, 16.634461444328644074630420675264, 17.05869289163802420063911715859, 18.11586457254758293796627263611, 18.6449883271375161939096878528, 19.84535728102685751289122016431, 21.04849896273388532775053782411, 21.32479705648572883906335490528, 22.10979328088533754867747059233

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