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Properties

Label 2-570-285.284-c1-0-13
Degree $2$
Conductor $570$
Sign $0.778 + 0.628i$
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 + (−1.41 + i)3-s − 4-s + (−1.73 + 1.41i)5-s + (1 + 1.41i)6-s − 2.44i·7-s + i·8-s + (1.00 − 2.82i)9-s + (1.41 + 1.73i)10-s + 1.41i·11-s + (1.41 − i)12-s − 4.24·13-s − 2.44·14-s + (1.03 − 3.73i)15-s + 16-s + 6.92·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.816 + 0.577i)3-s − 0.5·4-s + (−0.774 + 0.632i)5-s + (0.408 + 0.577i)6-s − 0.925i·7-s + 0.353i·8-s + (0.333 − 0.942i)9-s + (0.447 + 0.547i)10-s + 0.426i·11-s + (0.408 − 0.288i)12-s − 1.17·13-s − 0.654·14-s + (0.267 − 0.963i)15-s + 0.250·16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.778 + 0.628i$
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: \((2,\ 570,\ (\ :1/2),\ 0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809325 - 0.285940i\)
\(L(\frac12)\) \(\approx\) \(0.809325 - 0.285940i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + 7.34iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 2.44T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.59555086548566471382455631184, −10.05774879266278262339749804414, −9.390780802463900415880071792575, −7.65995044243845704150285997552, −7.31915108487835809235125666066, −5.88436842307765057757037177779, −4.75020795196865043886193082713, −3.95535799108867139064277337333, −2.97401590229859143539686600756, −0.789438381233177665965792283585, 0.966176085487300710750832794509, 3.03858777808618800227591470711, 4.78682765682837991126848067913, 5.28157993116816970449625341151, 6.20694301121805943335323396438, 7.45956079957820625990224596941, 7.80940937483132143668893384871, 8.925492098757938793347713248534, 9.783980290973786767060688360493, 11.03976974283268020005006569680

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