Properties

Label 2-570-15.2-c1-0-20
Degree $2$
Conductor $570$
Sign $0.695 + 0.718i$
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  + (0.707 − 0.707i)2-s + (1.71 + 0.274i)3-s − 1.00i·4-s + (−2.05 − 0.893i)5-s + (1.40 − 1.01i)6-s + (2.76 + 2.76i)7-s + (−0.707 − 0.707i)8-s + (2.84 + 0.940i)9-s + (−2.08 + 0.818i)10-s − 4.37i·11-s + (0.274 − 1.71i)12-s + (−0.0858 + 0.0858i)13-s + 3.91·14-s + (−3.26 − 2.09i)15-s − 1.00·16-s + (4.90 − 4.90i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.987 + 0.158i)3-s − 0.500i·4-s + (−0.916 − 0.399i)5-s + (0.573 − 0.414i)6-s + (1.04 + 1.04i)7-s + (−0.250 − 0.250i)8-s + (0.949 + 0.313i)9-s + (−0.658 + 0.258i)10-s − 1.32i·11-s + (0.0793 − 0.493i)12-s + (−0.0238 + 0.0238i)13-s + 1.04·14-s + (−0.841 − 0.539i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33444 - 0.989862i\)
\(L(\frac12)\) \(\approx\) \(2.33444 - 0.989862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.71 - 0.274i)T \)
5 \( 1 + (2.05 + 0.893i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-2.76 - 2.76i)T + 7iT^{2} \)
11 \( 1 + 4.37iT - 11T^{2} \)
13 \( 1 + (0.0858 - 0.0858i)T - 13iT^{2} \)
17 \( 1 + (-4.90 + 4.90i)T - 17iT^{2} \)
23 \( 1 + (-4.04 - 4.04i)T + 23iT^{2} \)
29 \( 1 + 7.73T + 29T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 + (1.15 + 1.15i)T + 37iT^{2} \)
41 \( 1 + 3.99iT - 41T^{2} \)
43 \( 1 + (2.22 - 2.22i)T - 43iT^{2} \)
47 \( 1 + (9.37 - 9.37i)T - 47iT^{2} \)
53 \( 1 + (-1.61 - 1.61i)T + 53iT^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + (9.57 + 9.57i)T + 67iT^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (3.04 - 3.04i)T - 73iT^{2} \)
79 \( 1 - 1.24iT - 79T^{2} \)
83 \( 1 + (-9.27 - 9.27i)T + 83iT^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (1.05 + 1.05i)T + 97iT^{2} \)
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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

−11.01518180891995053730347024355, −9.501806782441309745261236598295, −8.925452447727962208046139610348, −8.065363067502712446413271385260, −7.39248298097016605078270835958, −5.57286591569422142239670583547, −4.94967850217664821780602697438, −3.66198749682503761870615793973, −2.94437682050826670552354692877, −1.44211815298456770253784565888, 1.74777736659390905694549753232, 3.38458837201971509234108888781, 4.13853300079050567586985872658, 4.95003477063023334495197826216, 6.72491383694653415262912662366, 7.47313142919455508045434891400, 7.82997524433880292734781242273, 8.741636157753737972011611674669, 10.05035747775627477742158487049, 10.80168194624770534854249412948

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