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Properties

Label 1-1021-1021.153-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.825 - 0.564i$
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  + (−0.786 − 0.617i)2-s + (0.932 + 0.361i)3-s + (0.237 + 0.971i)4-s + (0.489 − 0.872i)5-s + (−0.510 − 0.859i)6-s + (−0.0184 + 0.999i)7-s + (0.412 − 0.911i)8-s + (0.739 + 0.673i)9-s + (−0.923 + 0.384i)10-s + (0.999 − 0.0246i)11-s + (−0.128 + 0.991i)12-s + (0.273 − 0.961i)13-s + (0.631 − 0.775i)14-s + (0.771 − 0.636i)15-s + (−0.886 + 0.462i)16-s + (0.801 − 0.597i)17-s + ⋯
L(s)  = 1  + (−0.786 − 0.617i)2-s + (0.932 + 0.361i)3-s + (0.237 + 0.971i)4-s + (0.489 − 0.872i)5-s + (−0.510 − 0.859i)6-s + (−0.0184 + 0.999i)7-s + (0.412 − 0.911i)8-s + (0.739 + 0.673i)9-s + (−0.923 + 0.384i)10-s + (0.999 − 0.0246i)11-s + (−0.128 + 0.991i)12-s + (0.273 − 0.961i)13-s + (0.631 − 0.775i)14-s + (0.771 − 0.636i)15-s + (−0.886 + 0.462i)16-s + (0.801 − 0.597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.825 - 0.564i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.825 - 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.749302672 - 0.5408994382i\)
\(L(\frac12)\) \(\approx\) \(1.749302672 - 0.5408994382i\)
\(L(1)\) \(\approx\) \(1.218996173 - 0.2514304644i\)
\(L(1)\) \(\approx\) \(1.218996173 - 0.2514304644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.786 - 0.617i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (0.489 - 0.872i)T \)
7 \( 1 + (-0.0184 + 0.999i)T \)
11 \( 1 + (0.999 - 0.0246i)T \)
13 \( 1 + (0.273 - 0.961i)T \)
17 \( 1 + (0.801 - 0.597i)T \)
19 \( 1 + (0.952 + 0.303i)T \)
23 \( 1 + (0.843 - 0.536i)T \)
29 \( 1 + (-0.875 - 0.483i)T \)
31 \( 1 + (-0.949 - 0.314i)T \)
37 \( 1 + (-0.892 + 0.451i)T \)
41 \( 1 + (-0.952 - 0.303i)T \)
43 \( 1 + (0.730 + 0.682i)T \)
47 \( 1 + (0.355 + 0.934i)T \)
53 \( 1 + (0.366 - 0.930i)T \)
59 \( 1 + (0.456 - 0.889i)T \)
61 \( 1 + (-0.987 - 0.159i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (-0.412 + 0.911i)T \)
73 \( 1 + (0.631 + 0.775i)T \)
79 \( 1 + (0.687 - 0.726i)T \)
83 \( 1 + (-0.779 + 0.626i)T \)
89 \( 1 + (0.881 - 0.473i)T \)
97 \( 1 + (-0.992 - 0.122i)T \)
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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

−21.54233717360964917987646540399, −20.65109942191310191168764186768, −19.84101911479830788821043749658, −19.2348954226479415416713574095, −18.61018943614022487783024408448, −17.84328384183598253239683371354, −17.01432333204998799460198311529, −16.38848720538398510392882866099, −15.13132066269260824050379086036, −14.65038766838445809349468612956, −13.84903594338949474503969863240, −13.573599426169787495682022524432, −12.01727728600445632082779356221, −10.969938626738397928206584945629, −10.252086189565307435299605733395, −9.31232764274765991596282835261, −8.96175513631685784500939448562, −7.5802491462800711707941346715, −7.14878213220159195091087987342, −6.59041284569643584665084991910, −5.49170298990075454947086814546, −3.97700946809579565993198634300, −3.2196950974746808991679753230, −1.774477207618031463709698342385, −1.29474745483367832787129876614, 1.08774118822640972394848210526, 1.916469694274608050294943185, 2.9474248276259715110374633010, 3.64302979636644726520594216826, 4.869617126577163967076424262387, 5.797535836617166033517976927782, 7.212665123078865557739737460529, 8.14547701100024026622563693462, 8.765688411720416291366593430464, 9.46674794587066869888985918909, 9.837338802089634895286931229042, 11.03611141437445830215614379597, 12.04469913089583155075336248681, 12.67267954593145718963019795866, 13.44918736796293532832726107468, 14.41484284507669979898313732259, 15.37984615749604434916540651475, 16.182111887356716082740566405326, 16.80598269543341902372407036850, 17.76409044787385992162185387198, 18.60708998091156229644850561940, 19.18784445104887380907127299147, 20.16204695205368405340535068595, 20.64372680175714175438042006773, 21.12463910532102180954646099979

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