↑

Properties

Label 1-569-569.3-r1-0-0
Degree $1$
Conductor $569$
Sign $0.483 - 0.875i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.801 − 0.598i)2-s + (0.999 + 0.0110i)3-s + (0.283 − 0.958i)4-s + (−0.991 − 0.132i)5-s + (0.807 − 0.589i)6-s + (−0.773 + 0.633i)7-s + (−0.346 − 0.937i)8-s + (0.999 + 0.0221i)9-s + (−0.873 + 0.487i)10-s + (0.641 + 0.766i)11-s + (0.294 − 0.955i)12-s + (0.912 + 0.408i)13-s + (−0.240 + 0.970i)14-s + (−0.989 − 0.143i)15-s + (−0.839 − 0.544i)16-s + (0.945 + 0.325i)17-s + ⋯
L(s)  = 1  + (0.801 − 0.598i)2-s + (0.999 + 0.0110i)3-s + (0.283 − 0.958i)4-s + (−0.991 − 0.132i)5-s + (0.807 − 0.589i)6-s + (−0.773 + 0.633i)7-s + (−0.346 − 0.937i)8-s + (0.999 + 0.0221i)9-s + (−0.873 + 0.487i)10-s + (0.641 + 0.766i)11-s + (0.294 − 0.955i)12-s + (0.912 + 0.408i)13-s + (−0.240 + 0.970i)14-s + (−0.989 − 0.143i)15-s + (−0.839 − 0.544i)16-s + (0.945 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.483 - 0.875i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.483 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.660503261 - 2.161142368i\)
\(L(\frac12)\) \(\approx\) \(3.660503261 - 2.161142368i\)
\(L(1)\) \(\approx\) \(1.949322916 - 0.7393858922i\)
\(L(1)\) \(\approx\) \(1.949322916 - 0.7393858922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.801 - 0.598i)T \)
3 \( 1 + (0.999 + 0.0110i)T \)
5 \( 1 + (-0.991 - 0.132i)T \)
7 \( 1 + (-0.773 + 0.633i)T \)
11 \( 1 + (0.641 + 0.766i)T \)
13 \( 1 + (0.912 + 0.408i)T \)
17 \( 1 + (0.945 + 0.325i)T \)
19 \( 1 + (-0.878 - 0.477i)T \)
23 \( 1 + (0.624 - 0.780i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
31 \( 1 + (-0.0773 - 0.997i)T \)
37 \( 1 + (-0.0993 + 0.995i)T \)
41 \( 1 + (0.110 + 0.993i)T \)
43 \( 1 + (0.598 - 0.801i)T \)
47 \( 1 + (0.165 + 0.986i)T \)
53 \( 1 + (0.589 + 0.807i)T \)
59 \( 1 + (0.986 - 0.165i)T \)
61 \( 1 + (0.387 - 0.921i)T \)
67 \( 1 + (0.945 - 0.325i)T \)
71 \( 1 + (0.346 - 0.937i)T \)
73 \( 1 + (0.477 - 0.878i)T \)
79 \( 1 + (-0.304 - 0.952i)T \)
83 \( 1 + (-0.833 + 0.553i)T \)
89 \( 1 + (-0.997 - 0.0773i)T \)
97 \( 1 + (0.496 + 0.867i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.274784335363077441350342231863, −22.63385321266162318146028654119, −21.46411823686293061197069700501, −20.826852009439636785153372392967, −19.79414827894014043295252219459, −19.352743455589734111682580249095, −18.3257462908995308872699890107, −16.911368563202384258270840933876, −16.07609473172160511084128781551, −15.69676688067487652673122423084, −14.52217501017599533466729784141, −14.12160069787548284712797387056, −13.061271934953337083582701050966, −12.49958311567839043062040609051, −11.32907819989060754517412315765, −10.343555545936004674498909868217, −8.868355589683208799784977710350, −8.32771966359893068927507238756, −7.27932055318054263523310326717, −6.750299897713771894954871335747, −5.47508883483695187344130625506, −3.9131124263685360224668684120, −3.6972342626839071734276018552, −2.87676758796271144765858671352, −1.00820849957807252559344309433, 0.89286718163703444190835064881, 2.17569459676305443213184899735, 3.13484904643543732830795649789, 3.962394338539901814588003562425, 4.61372198799037878914388979193, 6.1983137575057873167069001824, 6.95528944811787209288608659918, 8.23367256380830947766082334640, 9.13014937458933372238152709963, 9.9113968775156560270057652799, 11.035532502844131245847510707960, 12.08611895993900638344643011138, 12.64898020957521470262976213254, 13.42817070067006215316120734216, 14.53468975769228604680268807632, 15.17599955027468277110788602928, 15.69772229128562582959175718386, 16.72976487429966612614056447283, 18.577092876019557191637073879202, 19.034291328677921513045387795869, 19.59983954252181685045000253820, 20.48937803258758633424570118766, 21.06530619498227130980493668904, 22.064290535826930937342387493646, 22.90097403117185391714523464746

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