Properties

Label 2-570-285.29-c1-0-11
Degree $2$
Conductor $570$
Sign $0.609 - 0.792i$
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  + (0.984 − 0.173i)2-s + (−1.43 + 0.974i)3-s + (0.939 − 0.342i)4-s + (1.35 + 1.77i)5-s + (−1.24 + 1.20i)6-s + (−1.28 − 0.739i)7-s + (0.866 − 0.5i)8-s + (1.10 − 2.79i)9-s + (1.64 + 1.51i)10-s + (2.81 − 1.62i)11-s + (−1.01 + 1.40i)12-s + (3.29 + 2.76i)13-s + (−1.38 − 0.505i)14-s + (−3.67 − 1.21i)15-s + (0.766 − 0.642i)16-s + (0.748 + 4.24i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.826 + 0.562i)3-s + (0.469 − 0.171i)4-s + (0.607 + 0.794i)5-s + (−0.506 + 0.493i)6-s + (−0.483 − 0.279i)7-s + (0.306 − 0.176i)8-s + (0.367 − 0.930i)9-s + (0.520 + 0.478i)10-s + (0.849 − 0.490i)11-s + (−0.292 + 0.405i)12-s + (0.913 + 0.766i)13-s + (−0.371 − 0.135i)14-s + (−0.949 − 0.314i)15-s + (0.191 − 0.160i)16-s + (0.181 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70293 + 0.838770i\)
\(L(\frac12)\) \(\approx\) \(1.70293 + 0.838770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (1.43 - 0.974i)T \)
5 \( 1 + (-1.35 - 1.77i)T \)
19 \( 1 + (-1.45 - 4.10i)T \)
good7 \( 1 + (1.28 + 0.739i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.81 + 1.62i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.29 - 2.76i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.748 - 4.24i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.23 - 2.63i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.791 + 4.48i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.19 - 1.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.11T + 37T^{2} \)
41 \( 1 + (1.02 - 0.862i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.81 - 7.73i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.06 + 11.6i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.503 + 1.38i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.10 + 11.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.53 + 3.10i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.31 + 13.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.07 + 0.757i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-7.62 - 9.08i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-4.55 - 5.43i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.29 - 2.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.44 - 2.89i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.63 + 9.26i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.03954968316784724599060894568, −10.00360168920113563887175447794, −9.673814369969194508022128083604, −8.165261679672271778033856599824, −6.51507153630478692281833823083, −6.38934735533636847113541768099, −5.51226269889424069393032007830, −3.93023192259009817502158199678, −3.58789533166274611305211252465, −1.65556570943739966142968683534, 1.09479866032038923317757131946, 2.56444848847842326354816395830, 4.22255743421841367172529815525, 5.19391705807394205796543541541, 5.98703045273034123395954798651, 6.64604494415281000572945696556, 7.71825404765691817469916536309, 8.856343134471542447970210265226, 9.820048498804390061167386029610, 10.77478030580128627382287194237

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