Properties

Label 2-570-95.64-c1-0-11
Degree $2$
Conductor $570$
Sign $0.742 + 0.669i$
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 + (2.05 − 0.873i)5-s + (−0.499 + 0.866i)6-s − 2.32i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.34 + 1.78i)10-s − 2.85·11-s − 0.999i·12-s + (5.20 + 3.00i)13-s + (1.16 + 2.01i)14-s + (1.34 − 1.78i)15-s + (−0.5 − 0.866i)16-s + (3.79 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.920 − 0.390i)5-s + (−0.204 + 0.353i)6-s − 0.877i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.425 + 0.564i)10-s − 0.859·11-s − 0.288i·12-s + (1.44 + 0.833i)13-s + (0.310 + 0.537i)14-s + (0.347 − 0.461i)15-s + (−0.125 − 0.216i)16-s + (0.920 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.742 + 0.669i$
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: \((2,\ 570,\ (\ :1/2),\ 0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39403 - 0.535966i\)
\(L(\frac12)\) \(\approx\) \(1.39403 - 0.535966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 + 0.873i)T \)
19 \( 1 + (2.63 + 3.47i)T \)
good7 \( 1 + 2.32iT - 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + (-5.20 - 3.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.79 + 2.19i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.36 + 2.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.41 - 7.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.685T + 31T^{2} \)
37 \( 1 + 7.79iT - 37T^{2} \)
41 \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 1.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.83 - 3.94i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.86 + 2.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.724 + 1.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.40 + 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.89 - 3.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.15 - 1.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.38 - 3.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.00 - 3.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 + (4.34 - 7.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.79 + 3.34i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.54111936086907313819061326670, −9.555917465500202133703367799771, −8.903184181751287866089086223919, −8.053941467248039539488701939383, −7.12438916073994765741600461602, −6.27818610332271147882782855339, −5.25674356025646729916916673932, −3.90711704883335121772727304016, −2.33194560803818436480698668085, −1.07846705924031717969005969352, 1.74742285424802915658289820323, 2.77786174369532756682261092203, 3.79965113326359950950662947601, 5.67239501522788238834664722808, 6.01166112530130951780292185737, 7.67953936681888708861538025075, 8.317557855480967627208389101303, 9.125556164972064848699322829201, 10.16597336670516927390057083812, 10.40241022657624409754774818253

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