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Properties

Label 2-570-95.64-c1-0-0
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: $\chi_{570} (349, \cdot )$
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

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

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