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Properties

Label 1-569-569.180-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.850 + 0.526i$
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  + (−0.862 + 0.506i)2-s + (−0.428 + 0.903i)3-s + (0.487 − 0.873i)4-s + (0.562 − 0.826i)5-s + (−0.0883 − 0.996i)6-s + (0.666 + 0.745i)7-s + (0.0221 + 0.999i)8-s + (−0.633 − 0.773i)9-s + (−0.0663 + 0.997i)10-s + (−0.387 + 0.921i)11-s + (0.580 + 0.814i)12-s + (0.448 + 0.894i)13-s + (−0.952 − 0.304i)14-s + (0.506 + 0.862i)15-s + (−0.525 − 0.850i)16-s + (−0.759 − 0.650i)17-s + ⋯
L(s)  = 1  + (−0.862 + 0.506i)2-s + (−0.428 + 0.903i)3-s + (0.487 − 0.873i)4-s + (0.562 − 0.826i)5-s + (−0.0883 − 0.996i)6-s + (0.666 + 0.745i)7-s + (0.0221 + 0.999i)8-s + (−0.633 − 0.773i)9-s + (−0.0663 + 0.997i)10-s + (−0.387 + 0.921i)11-s + (0.580 + 0.814i)12-s + (0.448 + 0.894i)13-s + (−0.952 − 0.304i)14-s + (0.506 + 0.862i)15-s + (−0.525 − 0.850i)16-s + (−0.759 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.850 + 0.526i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.850 + 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1891617864 + 0.6653054446i\)
\(L(\frac12)\) \(\approx\) \(0.1891617864 + 0.6653054446i\)
\(L(1)\) \(\approx\) \(0.5559496123 + 0.3721159496i\)
\(L(1)\) \(\approx\) \(0.5559496123 + 0.3721159496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.862 + 0.506i)T \)
3 \( 1 + (-0.428 + 0.903i)T \)
5 \( 1 + (0.562 - 0.826i)T \)
7 \( 1 + (0.666 + 0.745i)T \)
11 \( 1 + (-0.387 + 0.921i)T \)
13 \( 1 + (0.448 + 0.894i)T \)
17 \( 1 + (-0.759 - 0.650i)T \)
19 \( 1 + (-0.873 + 0.487i)T \)
23 \( 1 + (0.958 + 0.283i)T \)
29 \( 1 + (-0.650 - 0.759i)T \)
31 \( 1 + (-0.0442 + 0.999i)T \)
37 \( 1 + (-0.745 + 0.666i)T \)
41 \( 1 + (-0.283 + 0.958i)T \)
43 \( 1 + (0.862 + 0.506i)T \)
47 \( 1 + (-0.346 - 0.937i)T \)
53 \( 1 + (0.0883 + 0.996i)T \)
59 \( 1 + (0.346 + 0.937i)T \)
61 \( 1 + (0.975 + 0.219i)T \)
67 \( 1 + (-0.759 + 0.650i)T \)
71 \( 1 + (0.0221 - 0.999i)T \)
73 \( 1 + (0.873 - 0.487i)T \)
79 \( 1 + (-0.984 - 0.176i)T \)
83 \( 1 + (0.993 + 0.110i)T \)
89 \( 1 + (0.0442 + 0.999i)T \)
97 \( 1 + (-0.930 + 0.367i)T \)
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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

−22.79903189880097164998494047783, −22.0560363183087169820796522621, −21.126063010054698290041166818926, −20.33731354738363504921389857752, −19.20901169852004544457093877074, −18.8114269138138123821540543911, −17.74747160214301710074263741389, −17.50444788566713247595279152132, −16.678405151051545328910113439077, −15.42497723899548626713998784290, −14.255424553071959852995519512927, −13.20903003112911137021070177514, −12.85445405700053317257022464120, −11.23375539730171320825021792798, −10.95858010119006546044212995473, −10.41508786381538067567577845685, −8.85944670368188566564461827210, −8.09845071607851903897687721310, −7.19867262213927272355809504557, −6.479688458079071659112257396090, −5.4230380019570973408330809176, −3.72418375916361606770546693322, −2.5922968680493780637857968071, −1.72786937231350508659498801322, −0.50828592454351465285978033719, 1.433656172739992622243008532282, 2.41592652016498705242458826750, 4.45668493982738151511030612866, 5.04383572193163707112040987374, 5.90256875666311667896255155886, 6.872069073825818602783282292143, 8.304177720403780179968017068337, 8.99352152495942377963403657732, 9.54775364697869221884895940395, 10.52838404640376645500850382130, 11.41883758812939076351631437746, 12.223260950096344356809339879193, 13.59817853229159083423388761063, 14.75902413936213818222692778476, 15.328385526643404541133752081329, 16.18602465094906388687045240787, 16.89101783296952780465040642608, 17.66625458101129891983236952537, 18.2003048994531829617700383189, 19.35706977277413605049583179554, 20.62158583349429667030583383982, 20.853219240328889673629324934157, 21.71067819252614822071921195202, 22.97333001401706661721772251956, 23.675087707161690889968477877050

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