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Properties

Label 1-1045-1045.159-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.889 - 0.456i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s − 12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (−0.309 + 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.104 − 0.994i)24-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s − 12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (−0.309 + 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.104 − 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2717321809 + 1.123953993i\)
\(L(\frac12)\) \(\approx\) \(-0.2717321809 + 1.123953993i\)
\(L(1)\) \(\approx\) \(0.7575883745 + 0.8120632944i\)
\(L(1)\) \(\approx\) \(0.7575883745 + 0.8120632944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−20.81545888450599445416638862848, −20.24536011507259389777566145558, −19.728861247294421047259295417627, −18.99890905142055970661974260307, −18.312286802209790461273168327348, −17.35020076675370796683422640662, −16.53970424401363930624683433709, −15.31390599812259190663921006365, −14.43555604630141333912954899238, −13.9659280917952005329941509139, −12.98881099444818881754272786850, −12.71181788449689955874293443161, −11.45992909492251108931538693749, −10.70455230123629389943178254260, −9.692701572902388600647106318939, −9.32683982505674570310827241375, −8.29854238922147408152883550850, −7.37983366897937588827009930058, −6.57667821570935579176270370640, −5.00046223839079561612695240084, −4.208328200833034928911659921912, −3.47124899720782939069702596258, −2.486786445360931060470304601232, −1.74380477624891763722618500474, −0.38669548105053956392852348983, 1.78025366310093151121810821898, 2.92177571270713373873895292041, 3.71332328752044193792640089707, 4.84363633646451677624497901531, 5.505568519303662892288718178125, 6.67878270650223667898984663946, 7.40982083271334897592052772868, 8.29338525538519105731661108858, 9.02856138691752053813854070721, 9.536700556358400676370045560477, 10.51163971310752450129667164887, 11.94781143278071032819804712755, 12.91250730078414012258642098226, 13.320065917578135710516938199373, 14.4271746487289641956961914080, 15.0319727063882072593159484167, 15.49835712491208578236912373995, 16.26428497274755888376860322674, 17.26121399643991177861314641887, 18.04273235115583804007970647652, 18.94179492692482371100660739610, 19.44035882176086305203988316113, 20.427724261432704469729655644716, 21.457919582039328073561422581934, 22.08148207732845342139739254564

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