Properties

Label 2-570-95.49-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.0490 - 0.998i$
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 + (0.384 + 2.20i)5-s + (−0.499 − 0.866i)6-s + 2.51i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.768 − 2.09i)10-s + 2.88·11-s + 0.999i·12-s + (−4.03 + 2.32i)13-s + (1.25 − 2.18i)14-s + (−0.768 + 2.09i)15-s + (−0.5 + 0.866i)16-s + (−6.31 − 3.64i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.171 + 0.985i)5-s + (−0.204 − 0.353i)6-s + 0.951i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.242 − 0.664i)10-s + 0.869·11-s + 0.288i·12-s + (−1.11 + 0.645i)13-s + (0.336 − 0.582i)14-s + (−0.198 + 0.542i)15-s + (−0.125 + 0.216i)16-s + (−1.53 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0490 - 0.998i$
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: \((2,\ 570,\ (\ :1/2),\ -0.0490 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803362 + 0.843742i\)
\(L(\frac12)\) \(\approx\) \(0.803362 + 0.843742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.384 - 2.20i)T \)
19 \( 1 + (-4.11 - 1.43i)T \)
good7 \( 1 - 2.51iT - 7T^{2} \)
11 \( 1 - 2.88T + 11T^{2} \)
13 \( 1 + (4.03 - 2.32i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.31 + 3.64i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.48 + 0.855i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.20 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.22T + 31T^{2} \)
37 \( 1 - 3.26iT - 37T^{2} \)
41 \( 1 + (-4.48 + 7.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.84 - 3.37i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.98 - 2.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.38 + 5.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.73 + 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.50 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.23 - 5.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.38 - 9.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.2 - 7.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.11 + 7.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.14iT - 83T^{2} \)
89 \( 1 + (-2.37 - 4.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.98 - 1.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98186774610712773864430558967, −9.850881145578584332533718039361, −9.277356349159915422785488432551, −8.693313699171519054096992045185, −7.26135948564847193509056221384, −6.86202794678397991374763581225, −5.44537707796930227491984416567, −4.07127143491831289002077982874, −2.81937236431183300766929127523, −2.09256242134810151151352496589, 0.75813841244013705124521789867, 2.10513638000661952733863513143, 3.87199006726317981046084557811, 4.89289027342060229453143336795, 6.13444667749638940970772919942, 7.21003057808268470914819872178, 7.78625885180554925434200684643, 8.891158512320538397548870003001, 9.347695781805681976392198361627, 10.27658006379503282969128852682

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