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Properties

Label 2-570-95.12-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.989 + 0.144i$
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  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (−2.19 + 0.440i)5-s + (−0.499 + 0.866i)6-s + (−1.44 − 1.44i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.23 − 0.141i)10-s − 4.77·11-s + (−0.707 + 0.707i)12-s + (−6.13 + 1.64i)13-s + (−1.02 − 1.77i)14-s + (0.141 − 2.23i)15-s + (0.500 + 0.866i)16-s + (−0.624 + 2.33i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.980 + 0.197i)5-s + (−0.204 + 0.353i)6-s + (−0.547 − 0.547i)7-s + (0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.705 − 0.0448i)10-s − 1.44·11-s + (−0.204 + 0.204i)12-s + (−1.70 + 0.456i)13-s + (−0.273 − 0.474i)14-s + (0.0366 − 0.576i)15-s + (0.125 + 0.216i)16-s + (−0.151 + 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.989 + 0.144i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.989 + 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0333831 - 0.458678i\)
\(L(\frac12)\) \(\approx\) \(0.0333831 - 0.458678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.19 - 0.440i)T \)
19 \( 1 + (-4.35 + 0.0647i)T \)
good7 \( 1 + (1.44 + 1.44i)T + 7iT^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 + (6.13 - 1.64i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.624 - 2.33i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.619 - 2.31i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.138 + 0.240i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.20iT - 31T^{2} \)
37 \( 1 + (-2.38 + 2.38i)T - 37iT^{2} \)
41 \( 1 + (8.57 - 4.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.73 + 1.26i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.10 - 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.0540 + 0.0144i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.69 - 9.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.79 + 3.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.66 - 13.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.89 + 5.13i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.92 + 1.32i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0784 - 0.135i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.57 - 9.57i)T - 83iT^{2} \)
89 \( 1 + (-3.92 + 6.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.3 - 4.11i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.31824356420349932028203539794, −10.31716027465038411637783405030, −9.769302596555281715022511659775, −8.309225988479583316562719079944, −7.44758820357313154882260208695, −6.83039587131064472529803894474, −5.36876623571356122407232367122, −4.69098412922006777573401724785, −3.62913294723836633848546482190, −2.71303141251357909152475072698, 0.19492949327002167132650058511, 2.47882632559454743257440416796, 3.22954911345698847841791625401, 4.91773726866199869161970486376, 5.29316820418778598875480589507, 6.72974097280384615356982508790, 7.51881666129513722953713604911, 8.197774266031867390094422731013, 9.559695916535058143905866515609, 10.41428499512168733132915599627

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