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Properties

Label 2-570-95.8-c1-0-18
Degree $2$
Conductor $570$
Sign $-0.851 - 0.523i$
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 + (−1.02 − 1.98i)5-s + (0.499 + 0.866i)6-s + (−0.613 + 0.613i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (1.50 + 1.65i)10-s − 3.43·11-s + (−0.707 − 0.707i)12-s + (2.02 + 0.543i)13-s + (0.434 − 0.751i)14-s + (−1.65 + 1.50i)15-s + (0.500 − 0.866i)16-s + (−1.55 − 5.79i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.457 − 0.889i)5-s + (0.204 + 0.353i)6-s + (−0.232 + 0.232i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.474 + 0.523i)10-s − 1.03·11-s + (−0.204 − 0.204i)12-s + (0.562 + 0.150i)13-s + (0.116 − 0.200i)14-s + (−0.427 + 0.387i)15-s + (0.125 − 0.216i)16-s + (−0.376 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.851 - 0.523i$
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: \((2,\ 570,\ (\ :1/2),\ -0.851 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0147136 + 0.0519993i\)
\(L(\frac12)\) \(\approx\) \(0.0147136 + 0.0519993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 1.98i)T \)
19 \( 1 + (3.87 - 2.00i)T \)
good7 \( 1 + (0.613 - 0.613i)T - 7iT^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + (-2.02 - 0.543i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.55 + 5.79i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (0.621 - 2.31i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.59 - 7.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.09iT - 31T^{2} \)
37 \( 1 + (7.31 + 7.31i)T + 37iT^{2} \)
41 \( 1 + (6.91 + 3.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.61 + 0.968i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (7.08 + 1.89i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.31 + 0.887i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.68 - 6.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.34 - 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.38 + 5.18i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.85 + 3.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.4 - 3.07i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.509 - 0.509i)T + 83iT^{2} \)
89 \( 1 + (6.81 + 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.26 + 0.874i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.26111703559731739145320785686, −8.924027161650583601151367007382, −8.648589428604805618780767545110, −7.57253937718846141645733746984, −6.85764735901614493922210628552, −5.63568241031789438813581886376, −4.81935340121085525783053385557, −3.13796509994356509720068129344, −1.63049230991839707228302156326, −0.03746588001435536528141271809, 2.34378881750212084767751608946, 3.47488808217612115091953175044, 4.48324401452158808824747254042, 6.09165095852415785159433748608, 6.70388079774866354621548662812, 8.086255229405131495238061022238, 8.369788047543904601188784089900, 9.859544694503173863579390556088, 10.35896290842143162593123213826, 10.98808010019427858550997140118

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