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Properties

Label 1-1021-1021.718-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.993 - 0.113i$
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  + (0.961 + 0.273i)2-s + (−0.932 + 0.361i)3-s + (0.850 + 0.526i)4-s + (0.0922 + 0.995i)5-s + (−0.995 + 0.0922i)6-s + (0.798 − 0.602i)7-s + (0.673 + 0.739i)8-s + (0.739 − 0.673i)9-s + (−0.183 + 0.982i)10-s + (−0.982 + 0.183i)11-s + (−0.982 − 0.183i)12-s + (−0.961 + 0.273i)13-s + (0.932 − 0.361i)14-s + (−0.445 − 0.895i)15-s + (0.445 + 0.895i)16-s + (−0.0922 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.961 + 0.273i)2-s + (−0.932 + 0.361i)3-s + (0.850 + 0.526i)4-s + (0.0922 + 0.995i)5-s + (−0.995 + 0.0922i)6-s + (0.798 − 0.602i)7-s + (0.673 + 0.739i)8-s + (0.739 − 0.673i)9-s + (−0.183 + 0.982i)10-s + (−0.982 + 0.183i)11-s + (−0.982 − 0.183i)12-s + (−0.961 + 0.273i)13-s + (0.932 − 0.361i)14-s + (−0.445 − 0.895i)15-s + (0.445 + 0.895i)16-s + (−0.0922 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.993 - 0.113i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.993 - 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.690710611 - 0.1536922443i\)
\(L(\frac12)\) \(\approx\) \(2.690710611 - 0.1536922443i\)
\(L(1)\) \(\approx\) \(1.449483358 + 0.3883455982i\)
\(L(1)\) \(\approx\) \(1.449483358 + 0.3883455982i\)

Euler product

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

−21.50371626149764301871411222059, −20.93702479109043351924388394682, −20.046648516521458967708379716219, −19.13680870599372194447735370197, −18.33082156112794092870417036276, −17.33469426367322644880483173366, −16.78201650362954205875495074915, −15.71535315710816349381561985712, −15.30370470446327433304274156386, −14.096810461113064720369772324199, −13.28102371068850840080743197652, −12.49792687306839133004664563041, −12.13685910946727214331688247703, −11.29396812759434241113527314422, −10.464734753266350436644488341606, −9.61342009029425781917751786003, −8.10533715696227523913243787734, −7.60464393140621853902164585645, −6.26956304129889996457103289497, −5.44298405608155592550235675121, −5.08989218657557556651474568133, −4.30914630702922223926882785575, −2.82456210439780726012273264473, −1.7452618686781484299110255319, −1.05173124103160962892311696281, 0.471156454118023593463388294244, 2.15529789130204362720992555165, 2.97539377265102432649059773756, 4.18969013541060101213861295041, 4.94438705830466598737315681730, 5.45479141331678587165211118391, 6.849115168250597037689797944985, 7.03061000537336720080097294934, 7.97828465631482838762974805122, 9.63334203342841743446774798577, 10.60642448921870720084569602739, 10.973478993609773140736770550148, 11.84855607443699044946746689202, 12.50904860977909766198903897988, 13.812171786820489761174771264748, 14.110611217352020437406677477790, 15.28741715947244596778369212937, 15.59776615633773619566136532839, 16.62834617452166537649115415697, 17.47789623576620483971108000761, 17.922729397285358564841979934784, 18.96474134281818658485723028239, 20.189036031831707463414174439955, 20.96089937668686056267722073716, 21.52124208627961939406593991140

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