Properties

Label 2-570-95.64-c1-0-8
Degree $2$
Conductor $570$
Sign $0.983 - 0.178i$
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  + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (1.30 + 1.81i)5-s + (0.499 − 0.866i)6-s − 3i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.03 − 0.917i)10-s + 2·11-s + 0.999i·12-s + (−1.25 − 0.724i)13-s + (1.5 + 2.59i)14-s + (−2.03 − 0.917i)15-s + (−0.5 − 0.866i)16-s + (4.24 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.584 + 0.811i)5-s + (0.204 − 0.353i)6-s − 1.13i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.644 − 0.290i)10-s + 0.603·11-s + 0.288i·12-s + (−0.348 − 0.201i)13-s + (0.400 + 0.694i)14-s + (−0.526 − 0.236i)15-s + (−0.125 − 0.216i)16-s + (1.02 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04496 + 0.0942740i\)
\(L(\frac12)\) \(\approx\) \(1.04496 + 0.0942740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-1.30 - 1.81i)T \)
19 \( 1 + (-1 + 4.24i)T \)
good7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (1.25 + 0.724i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.24 + 2.44i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.67 + 4.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.89T + 31T^{2} \)
37 \( 1 - 9.44iT - 37T^{2} \)
41 \( 1 + (-1.22 - 2.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.71 + 2.72i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.31 - 4.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.1 - 6.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.89 + 6.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.98 + 1.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.22 - 7.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.81 - 1.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.39 + 2.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + (8.22 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.26 - 5.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.52884637740671785517521092899, −9.957101176595361059348646455926, −9.321137278288480554333162772725, −7.908282392948016683220132130618, −7.08023512671725296117365651049, −6.44619781583995496459398006128, −5.40766898614120270087026314711, −4.20140799789345866755404427896, −2.79481406502165989973807432770, −0.953929358926817826036942558078, 1.25678251670961016671837989348, 2.33821348194084421293385029238, 3.99783345304522377314605243596, 5.49384908742329454756027936195, 5.91881531608505229585605048421, 7.22787978470190755743240141420, 8.318823854579648938413719689481, 8.993881754212738176774364248708, 9.800942129050661281277459974394, 10.56630725497114564935355215658

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