Properties

Label 2-570-95.8-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.983 - 0.183i$
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 − 0.419i)5-s + (−0.499 − 0.866i)6-s + (−2.57 + 2.57i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−2.01 + 0.973i)10-s − 5.12·11-s + (0.707 + 0.707i)12-s + (−4.44 − 1.19i)13-s + (1.81 − 3.14i)14-s + (0.973 + 2.01i)15-s + (0.500 − 0.866i)16-s + (0.673 + 2.51i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (0.982 − 0.187i)5-s + (−0.204 − 0.353i)6-s + (−0.971 + 0.971i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.636 + 0.307i)10-s − 1.54·11-s + (0.204 + 0.204i)12-s + (−1.23 − 0.330i)13-s + (0.485 − 0.841i)14-s + (0.251 + 0.519i)15-s + (0.125 − 0.216i)16-s + (0.163 + 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.983 - 0.183i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.983 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471186 + 0.509996i\)
\(L(\frac12)\) \(\approx\) \(0.0471186 + 0.509996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 0.419i)T \)
19 \( 1 + (3.94 + 1.86i)T \)
good7 \( 1 + (2.57 - 2.57i)T - 7iT^{2} \)
11 \( 1 + 5.12T + 11T^{2} \)
13 \( 1 + (4.44 + 1.19i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.673 - 2.51i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.45 - 5.42i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.79 - 8.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.03iT - 31T^{2} \)
37 \( 1 + (-4.87 - 4.87i)T + 37iT^{2} \)
41 \( 1 + (5.16 + 2.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.264 - 0.0709i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.50 + 1.74i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.48 - 2.54i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.25 - 5.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.80 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.78 - 6.65i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.61 - 3.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.48 - 1.73i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.09 + 7.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.64 + 1.64i)T + 83iT^{2} \)
89 \( 1 + (-7.97 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.01 - 0.270i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.62077218166006468220119582362, −10.12384934253387803665024557333, −9.485525192650334571215788190832, −8.722412125440093036330887588848, −7.80940998699540868020062291230, −6.56313144079548793556576221213, −5.63590729753016469640087325101, −4.99158241552309927710742138660, −2.98560801841593335396224357596, −2.25741822995636417764526723086, 0.31560885308172857147822943722, 2.21438224106895458845464694212, 2.94579313194509325568570429407, 4.69789771651807154244273510288, 6.08031645318397090626716596988, 6.84746022540772347906885750229, 7.60022535631912367159263863615, 8.550469290518957253836142340905, 9.828001255877850580155212366744, 10.06569179378628852256632820397

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