Properties

Label 2-570-285.263-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.983 + 0.180i$
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  + (0.0871 + 0.996i)2-s + (0.604 + 1.62i)3-s + (−0.984 + 0.173i)4-s + (−0.885 + 2.05i)5-s + (−1.56 + 0.743i)6-s + (2.10 + 0.562i)7-s + (−0.258 − 0.965i)8-s + (−2.26 + 1.96i)9-s + (−2.12 − 0.703i)10-s + (−4.26 − 2.46i)11-s + (−0.876 − 1.49i)12-s + (1.55 + 3.33i)13-s + (−0.377 + 2.14i)14-s + (−3.86 − 0.196i)15-s + (0.939 − 0.342i)16-s + (−0.0644 − 0.737i)17-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (0.348 + 0.937i)3-s + (−0.492 + 0.0868i)4-s + (−0.395 + 0.918i)5-s + (−0.638 + 0.303i)6-s + (0.793 + 0.212i)7-s + (−0.0915 − 0.341i)8-s + (−0.756 + 0.653i)9-s + (−0.671 − 0.222i)10-s + (−1.28 − 0.742i)11-s + (−0.253 − 0.431i)12-s + (0.431 + 0.926i)13-s + (−0.100 + 0.572i)14-s + (−0.998 − 0.0507i)15-s + (0.234 − 0.0855i)16-s + (−0.0156 − 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.983 + 0.180i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114405 - 1.26078i\)
\(L(\frac12)\) \(\approx\) \(0.114405 - 1.26078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 0.996i)T \)
3 \( 1 + (-0.604 - 1.62i)T \)
5 \( 1 + (0.885 - 2.05i)T \)
19 \( 1 + (3.80 - 2.12i)T \)
good7 \( 1 + (-2.10 - 0.562i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.26 + 2.46i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.55 - 3.33i)T + (-8.35 + 9.95i)T^{2} \)
17 \( 1 + (0.0644 + 0.737i)T + (-16.7 + 2.95i)T^{2} \)
23 \( 1 + (-6.88 - 4.82i)T + (7.86 + 21.6i)T^{2} \)
29 \( 1 + (-5.94 + 4.99i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.68 + 2.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 - 2.59i)T - 37iT^{2} \)
41 \( 1 + (-1.88 - 5.18i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.20 + 0.841i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (3.40 + 0.297i)T + (46.2 + 8.16i)T^{2} \)
53 \( 1 + (1.78 + 1.25i)T + (18.1 + 49.8i)T^{2} \)
59 \( 1 + (-0.260 - 0.218i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.17 - 12.3i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.241 - 2.75i)T + (-65.9 - 11.6i)T^{2} \)
71 \( 1 + (-14.7 - 2.59i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (-6.92 - 3.22i)T + (46.9 + 55.9i)T^{2} \)
79 \( 1 + (-0.282 - 0.775i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.94 - 1.59i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.88 - 2.14i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.02 - 0.177i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.08591160798883379571181734341, −10.37426244860598334192860430525, −9.349723899405084493013681938827, −8.294645800792671330468497515957, −7.939285264253765440115083912015, −6.69434194409598218758630428454, −5.61536676220717037596668733978, −4.70575225705608002877844686665, −3.67744131470145228346981890060, −2.56088349619935267677901745527, 0.68212385662360040332805227523, 1.98300795416684464692435039845, 3.17983834508030053893798513808, 4.67530791160762629954492223837, 5.29555608112851443675679509423, 6.84045179618982661027548206822, 7.952136877152123775036644396976, 8.365666686006359871967255555142, 9.205020949186171205940415936501, 10.64441429126683795279493526166

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