Properties

Label 2-570-15.2-c1-0-17
Degree $2$
Conductor $570$
Sign $0.785 + 0.619i$
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  + (0.707 − 0.707i)2-s + (1.08 − 1.34i)3-s − 1.00i·4-s + (0.539 + 2.17i)5-s + (−0.182 − 1.72i)6-s + (2.86 + 2.86i)7-s + (−0.707 − 0.707i)8-s + (−0.629 − 2.93i)9-s + (1.91 + 1.15i)10-s + 4.60i·11-s + (−1.34 − 1.08i)12-s + (1.32 − 1.32i)13-s + 4.05·14-s + (3.51 + 1.63i)15-s − 1.00·16-s + (3.40 − 3.40i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.628 − 0.777i)3-s − 0.500i·4-s + (0.241 + 0.970i)5-s + (−0.0746 − 0.703i)6-s + (1.08 + 1.08i)7-s + (−0.250 − 0.250i)8-s + (−0.209 − 0.977i)9-s + (0.605 + 0.364i)10-s + 1.38i·11-s + (−0.388 − 0.314i)12-s + (0.368 − 0.368i)13-s + 1.08·14-s + (0.906 + 0.422i)15-s − 0.250·16-s + (0.825 − 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.785 + 0.619i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45608 - 0.852133i\)
\(L(\frac12)\) \(\approx\) \(2.45608 - 0.852133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.08 + 1.34i)T \)
5 \( 1 + (-0.539 - 2.17i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-2.86 - 2.86i)T + 7iT^{2} \)
11 \( 1 - 4.60iT - 11T^{2} \)
13 \( 1 + (-1.32 + 1.32i)T - 13iT^{2} \)
17 \( 1 + (-3.40 + 3.40i)T - 17iT^{2} \)
23 \( 1 + (2.00 + 2.00i)T + 23iT^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 + (4.28 + 4.28i)T + 37iT^{2} \)
41 \( 1 - 1.74iT - 41T^{2} \)
43 \( 1 + (-8.19 + 8.19i)T - 43iT^{2} \)
47 \( 1 + (5.22 - 5.22i)T - 47iT^{2} \)
53 \( 1 + (4.96 + 4.96i)T + 53iT^{2} \)
59 \( 1 - 5.08T + 59T^{2} \)
61 \( 1 + 0.446T + 61T^{2} \)
67 \( 1 + (-6.27 - 6.27i)T + 67iT^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 + (6.36 - 6.36i)T - 73iT^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + (-3.87 - 3.87i)T + 83iT^{2} \)
89 \( 1 - 2.07T + 89T^{2} \)
97 \( 1 + (-12.5 - 12.5i)T + 97iT^{2} \)
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

−10.83899660970284893160555112717, −9.746053748052139596424141543195, −9.002368938875564663343108854745, −7.79179969699816971127842836937, −7.17019722717343647604924514234, −5.99469876282860524073324086324, −5.11254973235245089321278717895, −3.63448883033544686376414197171, −2.44483144597876731709490753610, −1.83721424565808675016389601289, 1.55965727700314204931665143064, 3.54824103482353688222196000398, 4.15804978392357396965903819503, 5.18805152740703376935293443250, 5.92721155822431778313777826297, 7.58491140664547191902036972861, 8.191021650880122731917293066880, 8.815431981060034655207484727102, 9.878554671864890489822516160851, 10.90017914907171012093508485334

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