Properties

Label 2-570-95.8-c1-0-10
Degree $2$
Conductor $570$
Sign $0.132 + 0.991i$
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 + (0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (−2.02 − 0.939i)5-s + (−0.499 − 0.866i)6-s + (0.555 − 0.555i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (2.20 + 0.382i)10-s − 1.49·11-s + (0.707 + 0.707i)12-s + (1.35 + 0.362i)13-s + (−0.392 + 0.680i)14-s + (0.382 − 2.20i)15-s + (0.500 − 0.866i)16-s + (−1.14 − 4.28i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.907 − 0.420i)5-s + (−0.204 − 0.353i)6-s + (0.210 − 0.210i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.696 + 0.120i)10-s − 0.452·11-s + (0.204 + 0.204i)12-s + (0.374 + 0.100i)13-s + (−0.105 + 0.181i)14-s + (0.0987 − 0.568i)15-s + (0.125 − 0.216i)16-s + (−0.278 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440643 - 0.385512i\)
\(L(\frac12)\) \(\approx\) \(0.440643 - 0.385512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (2.02 + 0.939i)T \)
19 \( 1 + (2.08 + 3.82i)T \)
good7 \( 1 + (-0.555 + 0.555i)T - 7iT^{2} \)
11 \( 1 + 1.49T + 11T^{2} \)
13 \( 1 + (-1.35 - 0.362i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.14 + 4.28i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (0.232 - 0.867i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.46 + 7.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.39iT - 31T^{2} \)
37 \( 1 + (1.66 + 1.66i)T + 37iT^{2} \)
41 \( 1 + (4.17 + 2.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.74 + 1.27i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-11.7 - 3.16i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.07 - 0.555i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.90 + 3.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.36 + 9.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.000672 - 0.00251i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (11.8 + 6.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.71 - 0.459i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.131 + 0.227i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.84 + 9.84i)T + 83iT^{2} \)
89 \( 1 + (-6.69 - 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.96 + 1.86i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.58119480524504528080520732401, −9.404040039896338420317733843129, −8.920726802090868693332797280994, −7.84900801290047383037047159517, −7.35952555876895467374161046462, −5.97607120451947700746923819627, −4.80494060225080340831902883720, −3.92376237005508683355811026502, −2.46964369315990085659087238591, −0.41021943084484318924577667345, 1.57841077006627663233434573289, 2.97256849055699987609183291613, 4.01703935907714116843210074362, 5.61885818060729212706795056190, 6.75980643208449097165788162311, 7.47121931289873801815249314375, 8.448780771450238687910154836075, 8.755448659460088159934425911168, 10.42791792124475636915361485104, 10.69355736098292863130284762026

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