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Properties

Label 2-570-285.68-c1-0-33
Degree $2$
Conductor $570$
Sign $-0.736 + 0.676i$
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 + (0.158 + 1.72i)3-s + (0.866 − 0.499i)4-s + (0.448 − 2.19i)5-s + (−0.599 − 1.62i)6-s + (−2.12 + 2.12i)7-s + (−0.707 + 0.707i)8-s + (−2.94 + 0.548i)9-s + (0.133 + 2.23i)10-s − 5.82i·11-s + (0.999 + 1.41i)12-s + (−0.732 + 2.73i)13-s + (1.5 − 2.59i)14-s + (3.84 + 0.425i)15-s + (0.500 − 0.866i)16-s + (−6.02 + 1.61i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.0917 + 0.995i)3-s + (0.433 − 0.249i)4-s + (0.200 − 0.979i)5-s + (−0.244 − 0.663i)6-s + (−0.801 + 0.801i)7-s + (−0.249 + 0.249i)8-s + (−0.983 + 0.182i)9-s + (0.0423 + 0.705i)10-s − 1.75i·11-s + (0.288 + 0.408i)12-s + (−0.203 + 0.757i)13-s + (0.400 − 0.694i)14-s + (0.993 + 0.109i)15-s + (0.125 − 0.216i)16-s + (−1.46 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.736 + 0.676i$
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: \((2,\ 570,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0117883 - 0.0302429i\)
\(L(\frac12)\) \(\approx\) \(0.0117883 - 0.0302429i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.158 - 1.72i)T \)
5 \( 1 + (-0.448 + 2.19i)T \)
19 \( 1 + (4.33 + 0.5i)T \)
good7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (0.732 - 2.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (6.02 - 1.61i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (2.89 + 0.776i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.41 - 7.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.09 - 1.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.71 + 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.23 + 1.93i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.82 + 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.46 - 1.46i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.6 + 6.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.49 + 2.00i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.08 + 5.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 3i)T - 83iT^{2} \)
89 \( 1 + (-3.08 - 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.643 - 2.40i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.26268060722480953017557222964, −9.236604982924144940466075303494, −8.714990433086006242933255851499, −8.447859057502966115192088445970, −6.52042260698161796409875502457, −5.89880919058863313184174021025, −4.85225033675586293189887480513, −3.62647776918225261861986306600, −2.27224191956580121758689906776, −0.02049543277553428425949683741, 1.99606648072367873337238392905, 2.82278030401197105094768688030, 4.27786450608560827561813851672, 6.16867611998951763000597520861, 6.81754235570661925254090647039, 7.36336440412131703794055524497, 8.235574570394535858702036170859, 9.569195417221703917676746994288, 10.09929469721266527108608838761, 10.94074701679812576663115597376

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