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Properties

Label 1-1021-1021.316-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.188 + 0.982i$
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  + (−0.343 + 0.938i)2-s + (0.932 + 0.361i)3-s + (−0.763 − 0.645i)4-s + (−0.659 + 0.751i)5-s + (−0.659 + 0.751i)6-s + (0.956 − 0.291i)7-s + (0.869 − 0.494i)8-s + (0.739 + 0.673i)9-s + (−0.478 − 0.878i)10-s + (−0.128 − 0.991i)11-s + (−0.478 − 0.878i)12-s + (−0.273 + 0.961i)13-s + (−0.0554 + 0.998i)14-s + (−0.886 + 0.462i)15-s + (0.165 + 0.986i)16-s + (0.975 − 0.219i)17-s + ⋯
L(s)  = 1  + (−0.343 + 0.938i)2-s + (0.932 + 0.361i)3-s + (−0.763 − 0.645i)4-s + (−0.659 + 0.751i)5-s + (−0.659 + 0.751i)6-s + (0.956 − 0.291i)7-s + (0.869 − 0.494i)8-s + (0.739 + 0.673i)9-s + (−0.478 − 0.878i)10-s + (−0.128 − 0.991i)11-s + (−0.478 − 0.878i)12-s + (−0.273 + 0.961i)13-s + (−0.0554 + 0.998i)14-s + (−0.886 + 0.462i)15-s + (0.165 + 0.986i)16-s + (0.975 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344258848 + 1.110573869i\)
\(L(\frac12)\) \(\approx\) \(1.344258848 + 1.110573869i\)
\(L(1)\) \(\approx\) \(1.050247043 + 0.6276473461i\)
\(L(1)\) \(\approx\) \(1.050247043 + 0.6276473461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.343 + 0.938i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (-0.659 + 0.751i)T \)
7 \( 1 + (0.956 - 0.291i)T \)
11 \( 1 + (-0.128 - 0.991i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (0.975 - 0.219i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (0.165 - 0.986i)T \)
29 \( 1 + (0.956 + 0.291i)T \)
31 \( 1 + (-0.993 - 0.110i)T \)
37 \( 1 + (0.631 - 0.775i)T \)
41 \( 1 + (0.739 - 0.673i)T \)
43 \( 1 + (0.0184 - 0.999i)T \)
47 \( 1 + (-0.0554 + 0.998i)T \)
53 \( 1 + (-0.713 + 0.700i)T \)
59 \( 1 + (-0.966 + 0.255i)T \)
61 \( 1 + (-0.0554 + 0.998i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.869 - 0.494i)T \)
73 \( 1 + (-0.0554 - 0.998i)T \)
79 \( 1 + (0.903 + 0.429i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (-0.602 + 0.798i)T \)
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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

−21.058504866645784585080548071407, −20.47961059444140323962242445004, −20.00420992551835477924861370255, −19.32191148262894650426782033449, −18.36991538833519656074336185695, −17.85319461511514964287399654280, −17.006199677133895316330575708331, −15.8480588338935011095306761028, −14.952331639289571011963029753901, −14.305394237090206735908249977444, −13.23870006257492650022214058711, −12.50781948301060218799940111519, −12.07991170139047738385786248880, −11.1725791748107601760141786004, −9.893964492549365156155590392706, −9.49197946459420612255776606103, −8.25724026396952958681558910783, −7.98251507879074783144310282333, −7.36623405368922353268489899166, −5.38998023387114037314010615162, −4.63545180575931530348740607884, −3.675950181871980596575237754, −2.857563019751083025280975090820, −1.68686853006056375437022056257, −1.10183797190316796143293478519, 0.9937539107074569254316663971, 2.42133678965441054775155009803, 3.54959508281538429981167873659, 4.38356861294267422416651917070, 5.188795453283937506467883913833, 6.4724007563957386220008437830, 7.507572898571632316841343690346, 7.75037840874651940140076557040, 8.75326132128863235045803105251, 9.374296045736198778712469847120, 10.59371796962038548591842807907, 10.97693991665799072894651877234, 12.25024262653497045903432993205, 13.69568121907081529281146221552, 14.21460908756860206475748008165, 14.55015820446310606320763678883, 15.477155073580261482302251198219, 16.20516655982748225521925151008, 16.79211477014087726443874298989, 18.08210415859767034953381988152, 18.60986850102730500658992893675, 19.29629519593145850699080491956, 19.9783057424121849466487028706, 21.058786224839515814225846463482, 21.7936119357454764156526919808

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