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Properties

Label 1-1021-1021.19-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.738 - 0.673i$
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 + (0.816 + 0.577i)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 + (−0.943 − 0.332i)10-s + (0.650 − 0.759i)11-s + (−0.982 + 0.183i)12-s + (0.961 + 0.273i)13-s + (0.932 + 0.361i)14-s + (−0.552 − 0.833i)15-s + (0.445 − 0.895i)16-s + (0.908 − 0.417i)17-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (−0.932 − 0.361i)3-s + (0.850 − 0.526i)4-s + (0.816 + 0.577i)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 + (−0.943 − 0.332i)10-s + (0.650 − 0.759i)11-s + (−0.982 + 0.183i)12-s + (0.961 + 0.273i)13-s + (0.932 + 0.361i)14-s + (−0.552 − 0.833i)15-s + (0.445 − 0.895i)16-s + (0.908 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.738 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351965809 - 0.5239927587i\)
\(L(\frac12)\) \(\approx\) \(1.351965809 - 0.5239927587i\)
\(L(1)\) \(\approx\) \(0.7500467746 - 0.07523564617i\)
\(L(1)\) \(\approx\) \(0.7500467746 - 0.07523564617i\)

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 \( 1 + (0.816 + 0.577i)T \)
7 \( 1 + (-0.798 - 0.602i)T \)
11 \( 1 + (0.650 - 0.759i)T \)
13 \( 1 + (0.961 + 0.273i)T \)
17 \( 1 + (0.908 - 0.417i)T \)
19 \( 1 + (0.976 - 0.213i)T \)
23 \( 1 + (0.552 - 0.833i)T \)
29 \( 1 + (0.389 - 0.920i)T \)
31 \( 1 + (0.976 + 0.213i)T \)
37 \( 1 + (0.626 + 0.779i)T \)
41 \( 1 + (-0.213 - 0.976i)T \)
43 \( 1 + (0.920 - 0.389i)T \)
47 \( 1 + (-0.153 - 0.988i)T \)
53 \( 1 + (-0.833 - 0.552i)T \)
59 \( 1 + (0.626 + 0.779i)T \)
61 \( 1 + (0.779 - 0.626i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.739 + 0.673i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (0.982 - 0.183i)T \)
83 \( 1 + (0.153 + 0.988i)T \)
89 \( 1 + (-0.0307 + 0.999i)T \)
97 \( 1 + (-0.920 + 0.389i)T \)
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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

−21.346721466988224152192651483275, −20.820659488631093125686739451923, −19.94978188953587454047209133428, −19.00981932759334101835114907358, −18.14515113075998814184703299117, −17.64128507167880649821204037762, −16.91945949294964802542049940942, −16.143401949190075464717251526685, −15.74420974300028377681112223709, −14.610732697117915789363594607758, −13.15149203257916157522389050698, −12.51525625474126587780002888573, −11.90927262686551825919823199499, −10.98863592716512650813588949884, −9.99511747300166460993914112359, −9.575278457378637534233086949601, −8.961408131670208197005764780822, −7.75634733842058748846049268106, −6.580703224962311595814811307683, −6.04590625152085904073088111984, −5.22528152413112400096569481886, −3.87119308797871765026164095127, −2.89306892276215285118454921652, −1.42955969324044386772934384651, −0.95765238446321857343938618422, 0.70729020194723512107776251099, 1.132865391382500560451991380453, 2.51551115002007299751204042280, 3.5732090740262930425230644451, 5.20647647596916154768951651878, 6.119512966626087834843205317, 6.53925323672946724699417126472, 7.20524166399512926719405368226, 8.30239844834967086055900557792, 9.468282006558525253891674750657, 10.02021767580018900640272628466, 10.82384943413051738569478602557, 11.43836233750298935045198582851, 12.35540642673033123624566740651, 13.7273367551151814539664538592, 13.887500142871131227128902417716, 15.32158016911586553619577884600, 16.295697301449642012422314137186, 16.652847732003900369710376380499, 17.41206565637188007252614891607, 18.14516680018187610907482564080, 18.94682761559730614340630762597, 19.196944165308393390500547423941, 20.54140125246070951928727022374, 21.222635145110678380176062938737

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