↑

Properties

Label 2-570-95.88-c1-0-3
Degree $2$
Conductor $570$
Sign $0.805 - 0.593i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.809 − 2.08i)5-s + (−0.499 + 0.866i)6-s + (−3.60 + 3.60i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−2.22 + 0.242i)10-s + 2.76·11-s + (0.707 + 0.707i)12-s + (1.37 + 5.13i)13-s + (2.54 + 4.41i)14-s + (0.242 + 2.22i)15-s + (0.500 + 0.866i)16-s + (−3.93 − 1.05i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.361 − 0.932i)5-s + (−0.204 + 0.353i)6-s + (−1.36 + 1.36i)7-s + (−0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.702 + 0.0765i)10-s + 0.832·11-s + (0.204 + 0.204i)12-s + (0.381 + 1.42i)13-s + (0.681 + 1.18i)14-s + (0.0625 + 0.573i)15-s + (0.125 + 0.216i)16-s + (−0.955 − 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.805 - 0.593i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.805 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690809 + 0.226942i\)
\(L(\frac12)\) \(\approx\) \(0.690809 + 0.226942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.809 + 2.08i)T \)
19 \( 1 + (-4.32 - 0.570i)T \)
good7 \( 1 + (3.60 - 3.60i)T - 7iT^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
13 \( 1 + (-1.37 - 5.13i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.93 + 1.05i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-6.48 + 1.73i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.54 - 2.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.810iT - 31T^{2} \)
37 \( 1 + (2.19 + 2.19i)T + 37iT^{2} \)
41 \( 1 + (9.40 - 5.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.71 - 10.1i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.53 - 9.46i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.18 - 4.41i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.60 - 4.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.92 - 5.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.00 - 0.536i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.86 + 2.80i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.08 - 11.5i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.78 - 6.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.75 + 3.75i)T + 83iT^{2} \)
89 \( 1 + (2.32 - 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.15 + 15.4i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.29358382337623161833802400420, −9.718061003995409766255618347526, −9.153771751405018134156630483640, −8.708759452025332595099665164912, −6.94678287106137492369260761083, −6.18744146942680186560473550515, −5.16885609157423205213809285318, −4.19821578797556246624046974366, −2.96863004599012570719409615418, −1.43488357868544663397235546583, 0.44990575742512230825707348298, 3.38399923628780575115689369283, 3.73636844509602674920714159445, 5.21146995542205675101689461460, 6.40800777562940995496867271612, 6.88398852035361254572443216672, 7.53920602463579895481205240398, 8.893524154222310315886101543125, 10.01281931363711507833109733849, 10.49937445590349779889521949708

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