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Properties

Label 1-1021-1021.3-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.787 - 0.616i$
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  + (0.602 + 0.798i)2-s + (−0.982 + 0.183i)3-s + (−0.273 + 0.961i)4-s + (0.739 + 0.673i)5-s + (−0.739 − 0.673i)6-s + (−0.445 + 0.895i)7-s + (−0.932 + 0.361i)8-s + (0.932 − 0.361i)9-s + (−0.0922 + 0.995i)10-s + (0.0922 + 0.995i)11-s + (0.0922 − 0.995i)12-s + (0.602 − 0.798i)13-s + (−0.982 + 0.183i)14-s + (−0.850 − 0.526i)15-s + (−0.850 − 0.526i)16-s + (0.739 + 0.673i)17-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s + (−0.982 + 0.183i)3-s + (−0.273 + 0.961i)4-s + (0.739 + 0.673i)5-s + (−0.739 − 0.673i)6-s + (−0.445 + 0.895i)7-s + (−0.932 + 0.361i)8-s + (0.932 − 0.361i)9-s + (−0.0922 + 0.995i)10-s + (0.0922 + 0.995i)11-s + (0.0922 − 0.995i)12-s + (0.602 − 0.798i)13-s + (−0.982 + 0.183i)14-s + (−0.850 − 0.526i)15-s + (−0.850 − 0.526i)16-s + (0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.787 - 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4056424009 + 1.176561403i\)
\(L(\frac12)\) \(\approx\) \(-0.4056424009 + 1.176561403i\)
\(L(1)\) \(\approx\) \(0.6085948277 + 0.8647927938i\)
\(L(1)\) \(\approx\) \(0.6085948277 + 0.8647927938i\)

Euler product

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

−21.29388920477039506546900500790, −20.57761006568392581542439927897, −19.64224200878646101370625169462, −18.75852238758974231761423222339, −18.2292946352833188948985004276, −17.06818544022419379383498147724, −16.51760606538696085178059597125, −15.92587481940214860189016846338, −14.37719363625535413779905812694, −13.71564421993494637375264828098, −13.16648301523734867548951021831, −12.39892929564630336905685353565, −11.58312250820103335577120220565, −10.823778123314346296998376578749, −10.07778523903332968680443840188, −9.421334506119188065374038178176, −8.25286760835079149060859384450, −6.77115509716267089858161387570, −6.08002768220776265063078521410, −5.514183831153468402560294757328, −4.36001005320929406038759192852, −3.85681366144348721690839021746, −2.355522775537916286559049210968, −1.29967448069339416308453807679, −0.51157618143944891926582393909, 1.7671708157131674163272711918, 2.96729949301507878894698936137, 3.93175181588260879738814621596, 5.0659408859701829067668444052, 5.79110445505606563538003160873, 6.28649839015019438432675415892, 7.0391089428037202205917715760, 8.101033743502302830677906236358, 9.29212110183818233898438936325, 10.0168183879586919020331383440, 10.94498927154908322224455525989, 11.924566443939839663926067529484, 12.8756083794723573738308579777, 13.000788323085012081158675880370, 14.56860115336763322192339718533, 14.93561282001696179430321715430, 15.79088965466553960559130738575, 16.47961323698951167196269264311, 17.42782304171110925722248805443, 18.01806564353068571182211628172, 18.38588306153506908154444131296, 19.73801024944127540117137563619, 21.10315049183870446609263322884, 21.59690874876251134608174845165, 22.11616510840385540064379832938

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