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Properties

Label 1-1021-1021.982-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.913 - 0.405i$
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  + (−0.673 − 0.739i)2-s + (−0.445 + 0.895i)3-s + (−0.0922 + 0.995i)4-s + (−0.273 − 0.961i)5-s + (0.961 − 0.273i)6-s + (0.361 + 0.932i)7-s + (0.798 − 0.602i)8-s + (−0.602 − 0.798i)9-s + (−0.526 + 0.850i)10-s + (−0.850 + 0.526i)11-s + (−0.850 − 0.526i)12-s + (0.673 − 0.739i)13-s + (0.445 − 0.895i)14-s + (0.982 + 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.273 + 0.961i)17-s + ⋯
L(s)  = 1  + (−0.673 − 0.739i)2-s + (−0.445 + 0.895i)3-s + (−0.0922 + 0.995i)4-s + (−0.273 − 0.961i)5-s + (0.961 − 0.273i)6-s + (0.361 + 0.932i)7-s + (0.798 − 0.602i)8-s + (−0.602 − 0.798i)9-s + (−0.526 + 0.850i)10-s + (−0.850 + 0.526i)11-s + (−0.850 − 0.526i)12-s + (0.673 − 0.739i)13-s + (0.445 − 0.895i)14-s + (0.982 + 0.183i)15-s + (−0.982 − 0.183i)16-s + (0.273 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.913 - 0.405i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (982, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.913 - 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01091988400 - 0.05149502434i\)
\(L(\frac12)\) \(\approx\) \(0.01091988400 - 0.05149502434i\)
\(L(1)\) \(\approx\) \(0.5592763483 + 0.006513429636i\)
\(L(1)\) \(\approx\) \(0.5592763483 + 0.006513429636i\)

Euler product

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

−22.07445375911377435241293689548, −20.725652539566465590058518967241, −19.89887410193639396468958104605, −19.02520258720999723051342803198, −18.47277409637670817648542624241, −17.970945272096135372352310901900, −17.22496412677803225741108179107, −16.18554156613051230042020680330, −15.8533617022867489249218071846, −14.494618073491378473531750926639, −13.80308534905546742849536720210, −13.55687131931524978377632922375, −11.92571216938867178128729020104, −11.10078289770277983769021102564, −10.72423970238601278828007504956, −9.71596116299148683639582735587, −8.44800469822505405663322486791, −7.6869104455118564544654547804, −7.172463855395289405247135409246, −6.48006296834704960751436870083, −5.59068192345310311443392364847, −4.61180353339551272531086281886, −3.16550193157879575412440789795, −1.962583741120110135511499163728, −0.87517095684588251578888981224, 0.01968839055920416616114426222, 1.21438010230514768231140981106, 2.32918207716851723003857129278, 3.54794409593843167466056648156, 4.2512706775211678777014867087, 5.337315917732213970641270709605, 5.88158704721676580081081114818, 7.77148674635169935620126019691, 8.22463672373538997388349757328, 9.101328595107570845062317518239, 9.8185400843239320422214409632, 10.584736930280639188522076493244, 11.41442211507814251921169453625, 12.267280752878689518779067980979, 12.593671657206493650974260331800, 13.770454142185945937654310154314, 15.25987543580571441820023755485, 15.65559025601908039206928269838, 16.417359415720436139321276055748, 17.27338260059631157830994516095, 17.91701655301662149547840509989, 18.61555557021414939639318224000, 19.70083271382648799153695613873, 20.56291011812642150631842717377, 20.88812711822212673695697726371

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