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Properties

Label 2-570-285.284-c1-0-32
Degree $2$
Conductor $570$
Sign $0.793 + 0.608i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 1.73·3-s − 4-s + (−1.41 − 1.73i)5-s + 1.73i·6-s − 2.44i·7-s − i·8-s + 2.99·9-s + (1.73 − 1.41i)10-s − 3.46i·11-s − 1.73·12-s + 2.44·14-s + (−2.44 − 2.99i)15-s + 16-s − 7.07·17-s + 2.99i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.00·3-s − 0.5·4-s + (−0.632 − 0.774i)5-s + 0.707i·6-s − 0.925i·7-s − 0.353i·8-s + 0.999·9-s + (0.547 − 0.447i)10-s − 1.04i·11-s − 0.500·12-s + 0.654·14-s + (−0.632 − 0.774i)15-s + 0.250·16-s − 1.71·17-s + 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52380 - 0.517348i\)
\(L(\frac12)\) \(\approx\) \(1.52380 - 0.517348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - 1.73T \)
5 \( 1 + (1.41 + 1.73i)T \)
19 \( 1 + (1 + 4.24i)T \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 - 4.89T + 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 - 9.79iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 7.34T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 3.46T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62472985303151179417057258272, −9.224026111488329341511826407172, −8.869867773286257383721108188836, −7.995783351860746151780628036808, −7.25868369194303960327617412154, −6.34150582939312509274589194159, −4.68037594095885346974226115929, −4.23199634649369632486044375344, −2.93952964025944464049125240323, −0.854929678458422935899191392745, 2.03533364005633532298823334083, 2.79610603177335466150996532040, 3.92855912681879237507712714236, 4.83561706022625781999354187517, 6.53346422778748062779402020723, 7.36484363032592496795422748367, 8.467085931739181244187458121253, 8.995691743643611649514947417268, 10.03011196696214507838995659399, 10.72066122148416771631333163404

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