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Properties

Label 1-569-569.131-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.492 - 0.870i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.154 − 0.988i)2-s + (0.132 + 0.991i)3-s + (−0.952 + 0.304i)4-s + (−0.0221 − 0.999i)5-s + (0.958 − 0.283i)6-s + (0.367 − 0.930i)7-s + (0.448 + 0.894i)8-s + (−0.964 + 0.262i)9-s + (−0.984 + 0.176i)10-s + (0.873 + 0.487i)11-s + (−0.428 − 0.903i)12-s + (−0.325 − 0.945i)13-s + (−0.975 − 0.219i)14-s + (0.988 − 0.154i)15-s + (0.814 − 0.580i)16-s + (0.666 − 0.745i)17-s + ⋯
L(s)  = 1  + (−0.154 − 0.988i)2-s + (0.132 + 0.991i)3-s + (−0.952 + 0.304i)4-s + (−0.0221 − 0.999i)5-s + (0.958 − 0.283i)6-s + (0.367 − 0.930i)7-s + (0.448 + 0.894i)8-s + (−0.964 + 0.262i)9-s + (−0.984 + 0.176i)10-s + (0.873 + 0.487i)11-s + (−0.428 − 0.903i)12-s + (−0.325 − 0.945i)13-s + (−0.975 − 0.219i)14-s + (0.988 − 0.154i)15-s + (0.814 − 0.580i)16-s + (0.666 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5402314516 - 0.9267037146i\)
\(L(\frac12)\) \(\approx\) \(0.5402314516 - 0.9267037146i\)
\(L(1)\) \(\approx\) \(0.8142827555 - 0.4603307835i\)
\(L(1)\) \(\approx\) \(0.8142827555 - 0.4603307835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.154 - 0.988i)T \)
3 \( 1 + (0.132 + 0.991i)T \)
5 \( 1 + (-0.0221 - 0.999i)T \)
7 \( 1 + (0.367 - 0.930i)T \)
11 \( 1 + (0.873 + 0.487i)T \)
13 \( 1 + (-0.325 - 0.945i)T \)
17 \( 1 + (0.666 - 0.745i)T \)
19 \( 1 + (-0.304 + 0.952i)T \)
23 \( 1 + (-0.970 + 0.240i)T \)
29 \( 1 + (0.745 - 0.666i)T \)
31 \( 1 + (0.801 - 0.598i)T \)
37 \( 1 + (-0.930 - 0.367i)T \)
41 \( 1 + (0.240 + 0.970i)T \)
43 \( 1 + (0.154 - 0.988i)T \)
47 \( 1 + (0.912 + 0.408i)T \)
53 \( 1 + (-0.958 + 0.283i)T \)
59 \( 1 + (-0.912 - 0.408i)T \)
61 \( 1 + (-0.0663 - 0.997i)T \)
67 \( 1 + (0.666 + 0.745i)T \)
71 \( 1 + (0.448 - 0.894i)T \)
73 \( 1 + (0.304 - 0.952i)T \)
79 \( 1 + (0.839 + 0.544i)T \)
83 \( 1 + (0.683 - 0.730i)T \)
89 \( 1 + (-0.801 - 0.598i)T \)
97 \( 1 + (-0.0442 - 0.999i)T \)
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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

−23.87019317562607299784543842253, −22.88448360089710225573989735092, −22.01817256901539212685902745140, −21.47968854392336760920838464974, −19.569424134905190872052564230240, −19.168577214446252120003958733209, −18.48492270289926426089058385161, −17.68977471574832649377172188118, −17.074926969529740954818122894102, −15.867670031085877477006466486579, −14.92060149571571426164303513981, −14.21262270426495384880878244416, −13.86028264301755934508016334670, −12.427760724723871816908635888413, −11.78663234029996183565530401548, −10.63819151519582307208083670835, −9.311129442533730114641046312242, −8.569746562988136627675410985481, −7.76988581054883427888818599909, −6.59046012118148675372006558562, −6.43117996917656350855883561326, −5.28223204698177436324852287399, −3.86169276654287139943260319789, −2.59598866771212339148266730235, −1.41338626381932768413172895466, 0.62574712179748830476151764328, 1.85901573318301220840645786907, 3.30497044990640772038133261002, 4.18270726044680285443620422257, 4.73250120021101927380825551162, 5.762361963775224091021421997516, 7.77918034723024345074526655446, 8.30877889782700806260563200594, 9.56411798315971983430931500611, 9.89079621325509275870543750569, 10.813526748665173478951017791792, 11.898830120900914169215566387712, 12.42416851824900161116914998702, 13.79505106476915661010805794287, 14.18537418350486261096352263054, 15.41474362222344786034250119397, 16.539445062015313537325825607436, 17.16982424166410747181556505161, 17.70583347200992169897734687731, 19.22837290068787367595011783347, 20.00483779246544331276056404141, 20.54210885034806466802023981231, 20.954355562770045435181191116393, 22.00011408125657301195686787183, 22.84107711117555469699522230826

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