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Properties

Label 2-570-15.2-c1-0-27
Degree $2$
Conductor $570$
Sign $-0.474 + 0.880i$
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  + (0.707 − 0.707i)2-s + (1.47 − 0.901i)3-s − 1.00i·4-s + (−1.61 + 1.54i)5-s + (0.408 − 1.68i)6-s + (−1.95 − 1.95i)7-s + (−0.707 − 0.707i)8-s + (1.37 − 2.66i)9-s + (−0.0553 + 2.23i)10-s − 3.67i·11-s + (−0.901 − 1.47i)12-s + (2.60 − 2.60i)13-s − 2.76·14-s + (−1.00 + 3.74i)15-s − 1.00·16-s + (−1.52 + 1.52i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.853 − 0.520i)3-s − 0.500i·4-s + (−0.724 + 0.689i)5-s + (0.166 − 0.687i)6-s + (−0.737 − 0.737i)7-s + (−0.250 − 0.250i)8-s + (0.457 − 0.888i)9-s + (−0.0175 + 0.706i)10-s − 1.10i·11-s + (−0.260 − 0.426i)12-s + (0.722 − 0.722i)13-s − 0.737·14-s + (−0.259 + 0.965i)15-s − 0.250·16-s + (−0.370 + 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00931 - 1.69075i\)
\(L(\frac12)\) \(\approx\) \(1.00931 - 1.69075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.47 + 0.901i)T \)
5 \( 1 + (1.61 - 1.54i)T \)
19 \( 1 + iT \)
good7 \( 1 + (1.95 + 1.95i)T + 7iT^{2} \)
11 \( 1 + 3.67iT - 11T^{2} \)
13 \( 1 + (-2.60 + 2.60i)T - 13iT^{2} \)
17 \( 1 + (1.52 - 1.52i)T - 17iT^{2} \)
23 \( 1 + (-0.483 - 0.483i)T + 23iT^{2} \)
29 \( 1 - 8.02T + 29T^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 + (-7.32 - 7.32i)T + 37iT^{2} \)
41 \( 1 - 5.46iT - 41T^{2} \)
43 \( 1 + (3.11 - 3.11i)T - 43iT^{2} \)
47 \( 1 + (3.44 - 3.44i)T - 47iT^{2} \)
53 \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 0.989T + 61T^{2} \)
67 \( 1 + (-8.05 - 8.05i)T + 67iT^{2} \)
71 \( 1 + 4.07iT - 71T^{2} \)
73 \( 1 + (-11.1 + 11.1i)T - 73iT^{2} \)
79 \( 1 - 7.15iT - 79T^{2} \)
83 \( 1 + (2.66 + 2.66i)T + 83iT^{2} \)
89 \( 1 + 1.05T + 89T^{2} \)
97 \( 1 + (-5.51 - 5.51i)T + 97iT^{2} \)
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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

−10.60341955726341384943889794893, −9.732535496496129743880228102506, −8.528199234740233697472151175235, −7.86621791897109000802879640270, −6.69887130430836465784345820321, −6.16996194975893442523683694741, −4.32410320658723415839917238787, −3.35171980206110780779577640378, −2.89936168754194465862803712290, −0.905225086769090714396627070943, 2.25549843357139624042387206885, 3.57825094622571060900795140791, 4.36976718911214446390622091930, 5.24400865132986400961061644521, 6.59159281693953621747172231854, 7.49980498728774906284281320770, 8.494038111400247881604682931126, 9.073268015804632996877552856544, 9.824364753724226860469803309567, 11.12319212143070390125632131607

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