Properties

Label 2-570-95.49-c1-0-19
Degree $2$
Conductor $570$
Sign $-0.882 + 0.470i$
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 + (−1.78 − 1.34i)5-s + (−0.499 − 0.866i)6-s − 2.32i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.873 − 2.05i)10-s − 2.85·11-s − 0.999i·12-s + (−5.20 + 3.00i)13-s + (1.16 − 2.01i)14-s + (0.873 + 2.05i)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 + (−0.798 − 0.601i)5-s + (−0.204 − 0.353i)6-s − 0.877i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.276 − 0.650i)10-s − 0.859·11-s − 0.288i·12-s + (−1.44 + 0.833i)13-s + (0.310 − 0.537i)14-s + (0.225 + 0.531i)15-s + (−0.125 + 0.216i)16-s + (−0.920 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.882 + 0.470i$
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: \((2,\ 570,\ (\ :1/2),\ -0.882 + 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0906262 - 0.362558i\)
\(L(\frac12)\) \(\approx\) \(0.0906262 - 0.362558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 + 1.34i)T \)
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

−10.64219419259198105274981862130, −9.479885027645854153529659823667, −8.307728651634619153062446450112, −7.35303194083497362942921126163, −7.00209746051939006599418474676, −5.59828292955051732459158137802, −4.63488863467027693664738892936, −4.06370978826786625845592025002, −2.32564100932489144329559499395, −0.16990736490730590929846155585, 2.43598437746529874560303087356, 3.29942295993276909581243830986, 4.74354878060473647264900455263, 5.26232707846982464390384976684, 6.50801063935509196687429023019, 7.35184932596714746920083721916, 8.458121128685401521521393638492, 9.596267159893068557125531683933, 10.60080078719645891191273108625, 11.08453849758011082352624622139

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