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Properties

Label 2-570-95.49-c1-0-7
Degree $2$
Conductor $570$
Sign $0.985 + 0.171i$
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 + (−2.22 + 0.223i)5-s + (−0.499 − 0.866i)6-s − 1.07i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (2.03 + 0.919i)10-s + 0.410·11-s + 0.999i·12-s + (3.30 − 1.91i)13-s + (−0.537 + 0.931i)14-s + (−2.03 − 0.919i)15-s + (−0.5 + 0.866i)16-s + (1.08 + 0.627i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.994 + 0.0998i)5-s + (−0.204 − 0.353i)6-s − 0.406i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.644 + 0.290i)10-s + 0.123·11-s + 0.288i·12-s + (0.917 − 0.529i)13-s + (−0.143 + 0.248i)14-s + (−0.526 − 0.237i)15-s + (−0.125 + 0.216i)16-s + (0.263 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.985 + 0.171i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.985 + 0.171i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15713 - 0.100220i\)
\(L(\frac12)\) \(\approx\) \(1.15713 - 0.100220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.223i)T \)
19 \( 1 + (-3.85 - 2.03i)T \)
good7 \( 1 + 1.07iT - 7T^{2} \)
11 \( 1 - 0.410T + 11T^{2} \)
13 \( 1 + (-3.30 + 1.91i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.08 - 0.627i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.23 + 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.18 - 2.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 - 2.08iT - 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.92 + 1.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.58 - 2.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.32 + 2.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.25 + 2.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.37 + 5.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.07 - 4.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.79 + 8.30i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.02 - 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (2.72 + 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60551141185742951828036642931, −9.959690590867787427665979860732, −8.825701561547745333120460470260, −8.178496994305128203204883722081, −7.48584194420902207272516637375, −6.43164910256176178513517473275, −4.83716590805997393710240925982, −3.69408823883211306758074550663, −3.00777728619063227715220536292, −1.08554276390479888542018662878, 1.10135484412473183657276632379, 2.81275512159402173379147110702, 3.99787236049471190177883146553, 5.29255023609879052606313481179, 6.56106541180135699634613244985, 7.31619526324655051615154032846, 8.202401102645538237666770845173, 8.821668506544212969679354858099, 9.590645094286625497549534820324, 10.77324048843380419234974533589

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