Properties

Label 1-1021-1021.670-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.333 - 0.942i$
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  + (0.895 − 0.445i)2-s + (−0.0922 + 0.995i)3-s + (0.602 − 0.798i)4-s + (0.932 − 0.361i)5-s + (0.361 + 0.932i)6-s + (0.526 − 0.850i)7-s + (0.183 − 0.982i)8-s + (−0.982 − 0.183i)9-s + (0.673 − 0.739i)10-s + (0.739 − 0.673i)11-s + (0.739 + 0.673i)12-s + (−0.895 − 0.445i)13-s + (0.0922 − 0.995i)14-s + (0.273 + 0.961i)15-s + (−0.273 − 0.961i)16-s + (−0.932 + 0.361i)17-s + ⋯
L(s)  = 1  + (0.895 − 0.445i)2-s + (−0.0922 + 0.995i)3-s + (0.602 − 0.798i)4-s + (0.932 − 0.361i)5-s + (0.361 + 0.932i)6-s + (0.526 − 0.850i)7-s + (0.183 − 0.982i)8-s + (−0.982 − 0.183i)9-s + (0.673 − 0.739i)10-s + (0.739 − 0.673i)11-s + (0.739 + 0.673i)12-s + (−0.895 − 0.445i)13-s + (0.0922 − 0.995i)14-s + (0.273 + 0.961i)15-s + (−0.273 − 0.961i)16-s + (−0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (670, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.369248479 - 3.349770689i\)
\(L(\frac12)\) \(\approx\) \(2.369248479 - 3.349770689i\)
\(L(1)\) \(\approx\) \(1.897099090 - 0.7058875335i\)
\(L(1)\) \(\approx\) \(1.897099090 - 0.7058875335i\)

Euler product

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

−21.9490393516353307902172944089, −21.1132233076919119277201636677, −20.10259911218118839219953808504, −19.408528871677678710343194698179, −18.19340697683984513496091139513, −17.63915933259429647251041633633, −17.23476221486145311958938864699, −16.069081152751362980655544175261, −15.037839405181404200894700476269, −14.24990513062411939361365894348, −14.02656385258292571979528859593, −12.85484813567159744581940804456, −12.33215614470518903779729405897, −11.59934713855543017077853110118, −10.769452073291416749934339749268, −9.263483050812582718715513844186, −8.63665291424057498244976948633, −7.42564615709280893713111569082, −6.71031288361511165603959183271, −6.27090162918588946990150760897, −5.13025959220196587321862698292, −4.61060779160928464553296180625, −2.788732753659403561586731946, −2.38592714171323582082137741887, −1.491420168149020585358007194463, 0.54819367284897745210477381534, 1.68271745751449792305618970915, 2.726682764768876810043950669440, 3.90859658672846590377146815134, 4.36601394602636507821197395171, 5.448121032188718271055088448590, 5.88953115975925521983595488613, 7.02264421667334666598617955937, 8.358768970882910087919615739390, 9.42379344878221050375396408826, 10.11722720046365089833403951794, 10.68986859565025414802567118470, 11.56970510385432589216787613654, 12.33980767242931255035330626839, 13.58373451267712595470100112697, 13.92920946967708936142539234230, 14.687452843109137335253250341532, 15.489257500132476561677373484515, 16.47742659161545025529050378356, 17.132653194674813302622482161952, 17.74380767954464682066827332498, 19.2454801813210972730558888686, 20.00587330017662951360067237344, 20.5383884887149641044234979278, 21.33315216925276387914456298504

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