Properties

Label 2-570-285.284-c1-0-31
Degree $2$
Conductor $570$
Sign $0.559 + 0.828i$
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  + i·2-s + (0.5 − 1.65i)3-s − 4-s + (1.91 + 1.15i)5-s + (1.65 + 0.5i)6-s − 3.21i·7-s i·8-s + (−2.5 − 1.65i)9-s + (−1.15 + 1.91i)10-s − 4.31i·11-s + (−0.5 + 1.65i)12-s − 3.31·13-s + 3.21·14-s + (2.87 − 2.59i)15-s + 16-s + 3.21·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.288 − 0.957i)3-s − 0.5·4-s + (0.855 + 0.518i)5-s + (0.677 + 0.204i)6-s − 1.21i·7-s − 0.353i·8-s + (−0.833 − 0.552i)9-s + (−0.366 + 0.604i)10-s − 1.30i·11-s + (−0.144 + 0.478i)12-s − 0.919·13-s + 0.860·14-s + (0.742 − 0.669i)15-s + 0.250·16-s + 0.780·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36892 - 0.727319i\)
\(L(\frac12)\) \(\approx\) \(1.36892 - 0.727319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.5 + 1.65i)T \)
5 \( 1 + (-1.91 - 1.15i)T \)
19 \( 1 + (4.31 + 0.605i)T \)
good7 \( 1 + 3.21iT - 7T^{2} \)
11 \( 1 + 4.31iT - 11T^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
23 \( 1 - 7.04T + 23T^{2} \)
29 \( 1 + 8.25T + 29T^{2} \)
31 \( 1 + 2.61iT - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 3.82iT - 43T^{2} \)
47 \( 1 - 8.86T + 47T^{2} \)
53 \( 1 - iT - 53T^{2} \)
59 \( 1 - 5.64T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 1.81iT - 73T^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 + 6.43T + 83T^{2} \)
89 \( 1 + 2.61T + 89T^{2} \)
97 \( 1 - 3.36T + 97T^{2} \)
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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

−10.62839843747414029970862921554, −9.528096378898595961454926296180, −8.732643142963682322308619115480, −7.54793776701561679018477057122, −7.16836714918167322698103097687, −6.17873245145297124364570736141, −5.43758100834088969365253709505, −3.80826456301963973685822028475, −2.59831180681730077909049536057, −0.874620487700127789595186109791, 2.02166679814079099484046829008, 2.78343948186860015235716336647, 4.34830857532347625125336907580, 5.12188648414183212959296784722, 5.84763064941362546848696784863, 7.52166898856162210942310686422, 8.816220291104302378137142787199, 9.256838209890321935467360807473, 9.880480280114892180893248021901, 10.66201199161312921998886303279

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