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Properties

Label 1-1021-1021.864-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.870 + 0.493i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (0.995 + 0.0922i)2-s + (0.602 + 0.798i)3-s + (0.982 + 0.183i)4-s + (−0.850 + 0.526i)5-s + (0.526 + 0.850i)6-s + (−0.673 − 0.739i)7-s + (0.961 + 0.273i)8-s + (−0.273 + 0.961i)9-s + (−0.895 + 0.445i)10-s + (0.445 − 0.895i)11-s + (0.445 + 0.895i)12-s + (−0.995 + 0.0922i)13-s + (−0.602 − 0.798i)14-s + (−0.932 − 0.361i)15-s + (0.932 + 0.361i)16-s + (0.850 − 0.526i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0922i)2-s + (0.602 + 0.798i)3-s + (0.982 + 0.183i)4-s + (−0.850 + 0.526i)5-s + (0.526 + 0.850i)6-s + (−0.673 − 0.739i)7-s + (0.961 + 0.273i)8-s + (−0.273 + 0.961i)9-s + (−0.895 + 0.445i)10-s + (0.445 − 0.895i)11-s + (0.445 + 0.895i)12-s + (−0.995 + 0.0922i)13-s + (−0.602 − 0.798i)14-s + (−0.932 − 0.361i)15-s + (0.932 + 0.361i)16-s + (0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.870 + 0.493i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (864, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.870 + 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.453916324 + 1.174299779i\)
\(L(\frac12)\) \(\approx\) \(4.453916324 + 1.174299779i\)
\(L(1)\) \(\approx\) \(2.065059284 + 0.5379965538i\)
\(L(1)\) \(\approx\) \(2.065059284 + 0.5379965538i\)

Euler product

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

−21.22566595708977884958003543311, −20.51026044132394459704973439310, −19.72973335486567482569382937992, −19.36083298950727334488414699042, −18.592293765764644122832304963993, −17.296540605559853908857998332664, −16.472801340345316770268142820505, −15.50509304717329873896647514634, −14.89254826251470632003037660990, −14.387840480179518596079720728637, −13.03382462764961647688970628760, −12.77288004156731543088110241824, −11.9473196059498470470571403271, −11.62565878153563173835070978204, −9.95472863132661915326664756539, −9.27541564601007800845940909600, −8.063374408657320474044529533094, −7.387121853839215793921604119996, −6.68928605774816971368173125141, −5.60514382448806107638332714114, −4.75237444939306723387861675121, −3.583438172591731709977994955615, −3.018104252068959110047635306729, −1.93656055962469177662378529265, −0.930491236129606882220295693, 0.75101589889473135102802717080, 2.65768072902859012568728626340, 3.17081837616741189198877126028, 3.87383724972541380732716646760, 4.66271049523039794969140278173, 5.6358459340649853051413999951, 6.8862799613290881121940621429, 7.42809841123876960969754660886, 8.3076575217287368224606643064, 9.62175209197452032953733237948, 10.28630727023491219581102589771, 11.281437274874561765396464606832, 11.75147569561682685162677532593, 12.94692650631934974336617817439, 13.795860221332139977846702344231, 14.40060144830185139127292580202, 15.0181614220475088839670249676, 15.892580207676532970303510704372, 16.46215737989774169938628604062, 17.02382047821601733876555550329, 18.822710894813223904001355923716, 19.44210978825407681927394021661, 19.99321976454460313070510917761, 20.70697292418895610755641958368, 21.663579594029502997894948950115

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