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Properties

Label 2-570-57.50-c1-0-14
Degree $2$
Conductor $570$
Sign $0.251 - 0.967i$
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  + (0.5 + 0.866i)2-s + (1.72 + 0.194i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (0.691 + 1.58i)6-s − 1.96·7-s − 0.999·8-s + (2.92 + 0.670i)9-s + (0.866 + 0.499i)10-s + 4.91i·11-s + (−1.02 + 1.39i)12-s + (1.73 + 1.00i)13-s + (−0.980 − 1.69i)14-s + (1.58 − 0.691i)15-s + (−0.5 − 0.866i)16-s + (3.39 − 1.96i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.993 + 0.112i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.282 + 0.648i)6-s − 0.741·7-s − 0.353·8-s + (0.974 + 0.223i)9-s + (0.273 + 0.158i)10-s + 1.48i·11-s + (−0.297 + 0.402i)12-s + (0.481 + 0.278i)13-s + (−0.261 − 0.453i)14-s + (0.409 − 0.178i)15-s + (−0.125 − 0.216i)16-s + (0.823 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89398 + 1.46405i\)
\(L(\frac12)\) \(\approx\) \(1.89398 + 1.46405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.72 - 0.194i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (3.86 - 2.01i)T \)
good7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 - 4.91iT - 11T^{2} \)
13 \( 1 + (-1.73 - 1.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.39 + 1.96i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.91 - 3.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.51 + 4.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.233iT - 31T^{2} \)
37 \( 1 + 7.19iT - 37T^{2} \)
41 \( 1 + (5.88 + 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.94 + 5.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.60 + 0.929i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 - 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.25 + 7.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.91 + 6.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.7 + 7.93i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.26 - 3.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.13 - 1.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.7 + 6.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (5.48 - 9.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.3 - 7.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.65305060659991713606939263166, −9.653205440282966023035250124651, −9.272734035058760861874831950192, −8.212069061155733467998512213007, −7.27697530419327362297345076270, −6.59584587891475136459850231820, −5.31099464798766548952670635585, −4.27666864313901294270296957542, −3.28958944861741259225041530637, −1.95499568150997337099322736902, 1.28344952641066381348845899935, 3.07476894899707741606139710489, 3.18266083862641716603188518212, 4.70007793294961375393464656832, 6.08619789123531629885313898146, 6.74299150676895006628615755377, 8.279687265791309722211192916969, 8.756244363062249662662293884446, 9.795618295142188682775853901455, 10.48432891438316160552150281548

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