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Properties

Label 2-570-285.68-c1-0-19
Degree $2$
Conductor $570$
Sign $0.849 + 0.528i$
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  + (−0.965 + 0.258i)2-s + (1.69 − 0.366i)3-s + (0.866 − 0.499i)4-s + (−2.17 + 0.537i)5-s + (−1.54 + 0.791i)6-s + (−1.07 + 1.07i)7-s + (−0.707 + 0.707i)8-s + (2.73 − 1.23i)9-s + (1.95 − 1.08i)10-s − 5.54i·11-s + (1.28 − 1.16i)12-s + (−0.0848 + 0.316i)13-s + (0.759 − 1.31i)14-s + (−3.47 + 1.70i)15-s + (0.500 − 0.866i)16-s + (6.12 − 1.64i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.977 − 0.211i)3-s + (0.433 − 0.249i)4-s + (−0.970 + 0.240i)5-s + (−0.628 + 0.323i)6-s + (−0.406 + 0.406i)7-s + (−0.249 + 0.249i)8-s + (0.910 − 0.413i)9-s + (0.618 − 0.341i)10-s − 1.67i·11-s + (0.370 − 0.335i)12-s + (−0.0235 + 0.0878i)13-s + (0.203 − 0.351i)14-s + (−0.897 + 0.440i)15-s + (0.125 − 0.216i)16-s + (1.48 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.849 + 0.528i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.849 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21015 - 0.345794i\)
\(L(\frac12)\) \(\approx\) \(1.21015 - 0.345794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-1.69 + 0.366i)T \)
5 \( 1 + (2.17 - 0.537i)T \)
19 \( 1 + (-1.73 + 3.99i)T \)
good7 \( 1 + (1.07 - 1.07i)T - 7iT^{2} \)
11 \( 1 + 5.54iT - 11T^{2} \)
13 \( 1 + (0.0848 - 0.316i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.12 + 1.64i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-7.51 - 2.01i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.43 - 4.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.22T + 31T^{2} \)
37 \( 1 + (-3.78 + 3.78i)T - 37iT^{2} \)
41 \( 1 + (1.04 + 0.602i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.79 - 2.35i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.639 + 2.38i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.61 + 1.50i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.29 - 3.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.12 + 7.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.809 + 0.216i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.44 + 4.29i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.89 + 2.65i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.24 - 2.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.14 + 2.14i)T - 83iT^{2} \)
89 \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.39 - 8.94i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.65141906601158397152214752808, −9.391864813239716299662560208079, −8.903450375858714901609982061443, −8.036100462416065214417602090945, −7.36477662646395979189953878989, −6.47128661850490633915183264351, −5.14433255963057396432593988722, −3.32737324665020494921330647526, −3.03871701427699033465860735329, −0.935504941503973219979509999417, 1.42451013163214272355322745411, 3.01032999944395237054913097378, 3.88545471491220487523162010628, 4.93587600175613778427360743230, 6.81408952216824139580257975211, 7.57849089404979961834663959744, 8.065478882395619058789057082000, 9.119063947385166724556013047739, 9.944485321308861503437624360270, 10.38947280106003489418687053799

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