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Properties

Label 1-1021-1021.471-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.997 + 0.0700i$
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  + (−0.526 − 0.850i)2-s + (−0.739 − 0.673i)3-s + (−0.445 + 0.895i)4-s + (−0.982 − 0.183i)5-s + (−0.183 + 0.982i)6-s + (−0.961 + 0.273i)7-s + (0.995 − 0.0922i)8-s + (0.0922 + 0.995i)9-s + (0.361 + 0.932i)10-s + (0.932 + 0.361i)11-s + (0.932 − 0.361i)12-s + (0.526 − 0.850i)13-s + (0.739 + 0.673i)14-s + (0.602 + 0.798i)15-s + (−0.602 − 0.798i)16-s + (0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)2-s + (−0.739 − 0.673i)3-s + (−0.445 + 0.895i)4-s + (−0.982 − 0.183i)5-s + (−0.183 + 0.982i)6-s + (−0.961 + 0.273i)7-s + (0.995 − 0.0922i)8-s + (0.0922 + 0.995i)9-s + (0.361 + 0.932i)10-s + (0.932 + 0.361i)11-s + (0.932 − 0.361i)12-s + (0.526 − 0.850i)13-s + (0.739 + 0.673i)14-s + (0.602 + 0.798i)15-s + (−0.602 − 0.798i)16-s + (0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.997 + 0.0700i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.997 + 0.0700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6506306527 + 0.02282993296i\)
\(L(\frac12)\) \(\approx\) \(0.6506306527 + 0.02282993296i\)
\(L(1)\) \(\approx\) \(0.4877084726 - 0.2229313467i\)
\(L(1)\) \(\approx\) \(0.4877084726 - 0.2229313467i\)

Euler product

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

−21.69333658228033148188493471599, −20.37761790453874806645985624063, −19.73308775038782084205722466704, −18.80475616847969783661314712906, −18.38483094096860052987101539952, −17.08328291035500677627089245617, −16.62409714753654097452403736655, −15.93410935233151650165781435637, −15.60914527415794602965282009417, −14.37705195036047792769312531472, −13.93549124620435560463402552606, −12.377159570063596838419993086520, −11.76583139406545030909083118722, −10.79493884972611471230855303773, −10.099225348362749682479726151192, −9.23320777483819327228631329279, −8.57935410739037559246946981002, −7.30638511787378612087079873869, −6.68116273975620061662085528711, −6.00795112261626205040499840897, −4.927564859526204504830983467595, −3.967222112962715657622845452551, −3.36802295108183733435334525570, −1.14486390801539930996934504513, −0.32149653876621351735285012236, 0.76147350526838893813802651447, 1.439156046317961773218785505685, 2.957449474166002755974464407861, 3.6104172544311815792489600471, 4.7037781348386507642083679361, 5.86602489325065154495843763179, 6.84366764111969252705578025511, 7.78926674145129239579382882408, 8.28840717957950035523504424965, 9.57638721967743672538080112527, 10.15530230648959831927781668612, 11.28491647477802213072244042599, 11.94158490578287738455164632508, 12.290530037858886351219483186602, 13.14297295878345547060637319767, 13.93259141396144180371895135572, 15.462395234795735511335058138539, 16.1176252689477776093753267942, 16.92015592147088148882477155439, 17.58032634300473350833867070693, 18.57994281887335647686238668842, 19.004725878574483910057150079543, 19.80127403474596785206604416879, 20.233614542274663279355384272358, 21.45875530837850941460024307320

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