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Properties

Label 2-570-19.7-c3-0-21
Degree $2$
Conductor $570$
Sign $-0.436 - 0.899i$
Analytic cond. $33.6310$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s + (−3 + 5.19i)6-s + 12.4·7-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (−5 + 8.66i)10-s + 34.3·11-s − 12·12-s + (21.7 − 37.7i)13-s + (12.4 + 21.5i)14-s + (−7.50 + 12.9i)15-s + (−8 − 13.8i)16-s + (53.3 + 92.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + 0.673·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + 0.942·11-s − 0.288·12-s + (0.464 − 0.804i)13-s + (0.237 + 0.412i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.760 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ -0.436 - 0.899i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.202934703\)
\(L(\frac12)\) \(\approx\) \(3.202934703\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
19 \( 1 + (-81.9 + 11.6i)T \)
good7 \( 1 - 12.4T + 343T^{2} \)
11 \( 1 - 34.3T + 1.33e3T^{2} \)
13 \( 1 + (-21.7 + 37.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-53.3 - 92.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-3.53 + 6.12i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (93.1 - 161. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 166.T + 2.97e4T^{2} \)
37 \( 1 + 98.6T + 5.06e4T^{2} \)
41 \( 1 + (138. + 239. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-47.1 - 81.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (66.9 - 116. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-116. + 201. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (101. + 175. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (115. - 199. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-16.7 + 29.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-380. - 659. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-294. - 509. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (678. + 1.17e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 773.T + 5.71e5T^{2} \)
89 \( 1 + (498. - 862. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (28.7 + 49.7i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.55844874445697123260215395642, −9.704873537435041737853536496434, −8.663965474565132307051773588126, −8.014740847387304154297793524354, −7.00168075723711761472781543734, −5.92719319620837510607130352787, −5.15941912203602927681446264951, −3.93994960305124778699663623759, −3.15460403228478280247723626745, −1.43294179842841494333920820810, 0.941221978153201612701785401446, 1.78563813383805242847849408729, 3.12459565395511515748330159295, 4.28279320095921610477079682228, 5.26442707020897910192610934650, 6.32663326010342467896182249638, 7.38518086213171622048073725682, 8.392111779575714992340897862799, 9.324561905853752553526596266057, 9.876295529760074122091543889161

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