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Properties

Label 2-570-5.4-c1-0-8
Degree $2$
Conductor $570$
Sign $0.970 - 0.241i$
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 − i·3-s − 4-s + (2.17 − 0.539i)5-s + 6-s + 1.07i·7-s − i·8-s − 9-s + (0.539 + 2.17i)10-s + 3.41·11-s + i·12-s − 0.921i·13-s − 1.07·14-s + (−0.539 − 2.17i)15-s + 16-s − 0.340i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.970 − 0.241i)5-s + 0.408·6-s + 0.407i·7-s − 0.353i·8-s − 0.333·9-s + (0.170 + 0.686i)10-s + 1.03·11-s + 0.288i·12-s − 0.255i·13-s − 0.288·14-s + (−0.139 − 0.560i)15-s + 0.250·16-s − 0.0825i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69131 + 0.206968i\)
\(L(\frac12)\) \(\approx\) \(1.69131 + 0.206968i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-2.17 + 0.539i)T \)
19 \( 1 - T \)
good7 \( 1 - 1.07iT - 7T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + 0.921iT - 13T^{2} \)
17 \( 1 + 0.340iT - 17T^{2} \)
23 \( 1 + 0.921iT - 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 + 5.60iT - 37T^{2} \)
41 \( 1 + 1.07T + 41T^{2} \)
43 \( 1 - 0.738iT - 43T^{2} \)
47 \( 1 - 7.75iT - 47T^{2} \)
53 \( 1 - 2.68iT - 53T^{2} \)
59 \( 1 + 8.34T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4.68iT - 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.83iT - 73T^{2} \)
79 \( 1 + 4.73T + 79T^{2} \)
83 \( 1 - 11.0iT - 83T^{2} \)
89 \( 1 + 9.75T + 89T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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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

−10.64983898218509445969693029912, −9.589375674464845045970893414763, −8.973506822346086569116064769326, −8.125350023739014189971619480758, −7.00482405936751140089952448391, −6.22643467534479050420815535720, −5.55335048129087029829607774477, −4.38545624950981318904524002984, −2.74688008579784076951100849490, −1.26519813328542062943859763215, 1.40706244727869401407820124805, 2.81103339339700962416586161667, 3.93165019501242617773803477466, 4.91783412681503540662999688118, 6.04784676950927396276313986503, 6.95827494259426393390189946751, 8.417201594809582666886957864898, 9.253966258260568665685249635526, 9.949112969719379408725802515910, 10.52893622377821435770313961573

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