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Properties

Label 1-1021-1021.23-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.387 + 0.921i$
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  + (0.631 + 0.775i)2-s + (−0.850 + 0.526i)3-s + (−0.201 + 0.979i)4-s + (0.189 + 0.981i)5-s + (−0.945 − 0.326i)6-s + (−0.478 − 0.878i)7-s + (−0.886 + 0.462i)8-s + (0.445 − 0.895i)9-s + (−0.641 + 0.767i)10-s + (0.141 − 0.989i)11-s + (−0.343 − 0.938i)12-s + (0.932 + 0.361i)13-s + (0.378 − 0.925i)14-s + (−0.678 − 0.734i)15-s + (−0.918 − 0.395i)16-s + (0.856 + 0.515i)17-s + ⋯
L(s)  = 1  + (0.631 + 0.775i)2-s + (−0.850 + 0.526i)3-s + (−0.201 + 0.979i)4-s + (0.189 + 0.981i)5-s + (−0.945 − 0.326i)6-s + (−0.478 − 0.878i)7-s + (−0.886 + 0.462i)8-s + (0.445 − 0.895i)9-s + (−0.641 + 0.767i)10-s + (0.141 − 0.989i)11-s + (−0.343 − 0.938i)12-s + (0.932 + 0.361i)13-s + (0.378 − 0.925i)14-s + (−0.678 − 0.734i)15-s + (−0.918 − 0.395i)16-s + (0.856 + 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262740811 + 0.8385727063i\)
\(L(\frac12)\) \(\approx\) \(1.262740811 + 0.8385727063i\)
\(L(1)\) \(\approx\) \(0.9752580767 + 0.6114216545i\)
\(L(1)\) \(\approx\) \(0.9752580767 + 0.6114216545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.631 + 0.775i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.189 + 0.981i)T \)
7 \( 1 + (-0.478 - 0.878i)T \)
11 \( 1 + (0.141 - 0.989i)T \)
13 \( 1 + (0.932 + 0.361i)T \)
17 \( 1 + (0.856 + 0.515i)T \)
19 \( 1 + (0.552 - 0.833i)T \)
23 \( 1 + (0.116 - 0.993i)T \)
29 \( 1 + (-0.521 - 0.853i)T \)
31 \( 1 + (0.963 - 0.267i)T \)
37 \( 1 + (-0.434 - 0.900i)T \)
41 \( 1 + (0.552 - 0.833i)T \)
43 \( 1 + (0.923 - 0.384i)T \)
47 \( 1 + (0.612 + 0.790i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (0.722 - 0.691i)T \)
61 \( 1 + (-0.990 + 0.135i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.886 + 0.462i)T \)
73 \( 1 + (0.378 + 0.925i)T \)
79 \( 1 + (-0.999 + 0.0369i)T \)
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

−21.39564730001870595497988268981, −20.830092496018209301851686267, −19.9807297238663889242475000628, −19.16736725924558212166387369977, −18.31447614509828873524967159, −17.86882703155792633404132608346, −16.71107623071574521197969552143, −15.93542155102908498493308880678, −15.243525946803999794263049088191, −13.9745839357560486648709113995, −13.23520335046629581533470140513, −12.54556771428934362812172137098, −12.065015482499955426310266595418, −11.47580619557666640335254776726, −10.21722183039038376487285472522, −9.66045236043639197664632073369, −8.69885282297615551386615402662, −7.5071555385722592293948414476, −6.30656898616470333017373131594, −5.55784629084949953730660069770, −5.14569272379595081029973992138, −4.04127208344906003092346670666, −2.87764463150549455539667595771, −1.64798180663679913779893189392, −1.09083274017788692519594710273, 0.72246327548401402712188447524, 2.78920462378323243231734678979, 3.74286870847718452828872810778, 4.15179725212653354785276817483, 5.574943316005660689967154282321, 6.10882506123996719187674842473, 6.780439099081123813044036135133, 7.56244844799658206376917543059, 8.788813410269611927353053736734, 9.779790988388599216172994567497, 10.77583091018603413440746501674, 11.23565917896751031705107276808, 12.22917014542933905158596455293, 13.27841470861139151566829842309, 13.94520117038998021525565225818, 14.60679037972363945449341878855, 15.73697004454196867833115684073, 16.05454580483411023086574509763, 17.03260732604765580730405215401, 17.43664312482069070709457640173, 18.50102633023316905578113717080, 19.14990906096132303446328188782, 20.64674395768080491357146078389, 21.24168322632804974831793697642, 21.95661522009685753918119536559

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