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Properties

Label 2-570-19.11-c3-0-22
Degree $2$
Conductor $570$
Sign $0.846 - 0.532i$
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 + 28.6·7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (5 + 8.66i)10-s + 4.31·11-s − 12·12-s + (23 + 39.8i)13-s + (−28.6 + 49.5i)14-s + (−7.50 − 12.9i)15-s + (−8 + 13.8i)16-s + (−40.9 + 70.9i)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 + 1.54·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + 0.118·11-s − 0.288·12-s + (0.490 + 0.849i)13-s + (−0.546 + 0.946i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.584 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.321993979\)
\(L(\frac12)\) \(\approx\) \(2.321993979\)
\(L(\frac{5}{2})\) 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 \( 1 + (-63.4 - 53.2i)T \)
good7 \( 1 - 28.6T + 343T^{2} \)
11 \( 1 - 4.31T + 1.33e3T^{2} \)
13 \( 1 + (-23 - 39.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (40.9 - 70.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-64.4 - 111. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (35.3 + 61.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 189.T + 2.97e4T^{2} \)
37 \( 1 + 271.T + 5.06e4T^{2} \)
41 \( 1 + (-60.7 + 105. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (13.7 - 23.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (24.1 + 41.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (77.3 + 133. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-39.6 + 68.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-256. - 444. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-451. - 781. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (89.0 - 154. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-280. + 485. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-472. + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 107.T + 5.71e5T^{2} \)
89 \( 1 + (297. + 515. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (331. - 573. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.31835842806187287802110852608, −9.181984964033035503037691430761, −8.505739972757968797290619435555, −7.88317925993904083475652739511, −6.95822286981293337039918312090, −5.90513908129120528349470005135, −4.98378132437616436612615075180, −3.90491505791675420972133811775, −1.91493971110154579904168034785, −1.21785294547068484837626920303, 0.903270582174526713625775467691, 2.28005735691308386101124472357, 3.25302043094041052562975003617, 4.61535862907753880873288326729, 5.24378223244806136011236779545, 6.83627077681056343934685448502, 7.893040377120189276324567046678, 8.584135272472225342783941282320, 9.406529432483759832516325288847, 10.40591851596405464377999125189

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