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Properties

Label 1-1021-1021.145-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.722 + 0.691i$
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  + (0.982 + 0.183i)2-s + (−0.273 + 0.961i)3-s + (0.932 + 0.361i)4-s + (0.445 − 0.895i)5-s + (−0.445 + 0.895i)6-s + (−0.0922 + 0.995i)7-s + (0.850 + 0.526i)8-s + (−0.850 − 0.526i)9-s + (0.602 − 0.798i)10-s + (−0.602 − 0.798i)11-s + (−0.602 + 0.798i)12-s + (0.982 − 0.183i)13-s + (−0.273 + 0.961i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.982 + 0.183i)2-s + (−0.273 + 0.961i)3-s + (0.932 + 0.361i)4-s + (0.445 − 0.895i)5-s + (−0.445 + 0.895i)6-s + (−0.0922 + 0.995i)7-s + (0.850 + 0.526i)8-s + (−0.850 − 0.526i)9-s + (0.602 − 0.798i)10-s + (−0.602 − 0.798i)11-s + (−0.602 + 0.798i)12-s + (0.982 − 0.183i)13-s + (−0.273 + 0.961i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.722 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.765754692 + 1.110895425i\)
\(L(\frac12)\) \(\approx\) \(2.765754692 + 1.110895425i\)
\(L(1)\) \(\approx\) \(1.886276898 + 0.5678622247i\)
\(L(1)\) \(\approx\) \(1.886276898 + 0.5678622247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.982 + 0.183i)T \)
3 \( 1 + (-0.273 + 0.961i)T \)
5 \( 1 + (0.445 - 0.895i)T \)
7 \( 1 + (-0.0922 + 0.995i)T \)
11 \( 1 + (-0.602 - 0.798i)T \)
13 \( 1 + (0.982 - 0.183i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (0.850 - 0.526i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (0.0922 + 0.995i)T \)
31 \( 1 + (0.850 + 0.526i)T \)
37 \( 1 + (0.273 - 0.961i)T \)
41 \( 1 + (-0.850 + 0.526i)T \)
43 \( 1 + (-0.0922 + 0.995i)T \)
47 \( 1 + (-0.273 + 0.961i)T \)
53 \( 1 + (-0.739 + 0.673i)T \)
59 \( 1 + (0.273 - 0.961i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.850 - 0.526i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.602 + 0.798i)T \)
83 \( 1 + (-0.273 + 0.961i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (-0.0922 + 0.995i)T \)
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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

−21.59084488959354826148372724237, −20.71393013009539132303880850440, −20.14386814116597035801495317128, −18.995120171386681838428793021981, −18.69807884919044541315791009418, −17.488039590954428616283960103529, −17.00588974708280490848022130749, −15.83194542691102974479085498524, −14.9653071138529332145852980731, −14.13214448879148959544006356432, −13.380337731718977248196650501837, −13.18955992627773642018184909861, −11.94646742256716909589778250199, −11.32730669099326958372989024635, −10.417983187998441305301844544195, −9.94976418287233464497214310542, −8.07137937511574781329986710178, −7.35813601411884896752243177788, −6.67815106533567881101130486999, −5.98491136139627599616476508627, −5.131555656722302273267675131121, −3.83935662412924570743725397508, −3.07540554157777606509036029734, −1.9779493834402987913678267271, −1.22892611375309464152012443999, 1.14386606941730120657329476318, 2.80049083265893783347940527596, 3.20091892919541557556517516477, 4.59420038158164040800146673622, 5.18321685465423803379469787631, 5.72834102687212644909349499391, 6.515345070984363328396630927812, 8.073074410276633354711449075358, 8.78430580045558489287701614039, 9.575380923148889973344820986292, 10.72896350530490047015570799745, 11.40782348289650610228010119122, 12.20914940922892973765876754336, 13.01067965470792050194913498114, 13.81815198187512083460574202628, 14.59251756716812147038907042148, 15.675967878117526961509126514320, 16.05639755261443278903238449887, 16.48638872344648740925890362768, 17.61959083154360386040059851864, 18.443243315340824881539465484202, 19.74122965143411651944436048576, 20.65085045701893594662089515704, 21.04222336530414836669648165025, 21.64010952669947627920852479879

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