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Properties

Label 2-570-95.37-c1-0-17
Degree $2$
Conductor $570$
Sign $-0.982 + 0.185i$
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 + (0.707 + 0.707i)3-s − 1.00i·4-s + (−2.23 + 0.0685i)5-s + 1.00·6-s + (−2.16 − 2.16i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.53 + 1.62i)10-s − 5.68·11-s + (0.707 − 0.707i)12-s + (−3.92 − 3.92i)13-s − 3.06·14-s + (−1.62 − 1.53i)15-s − 1.00·16-s + (4.99 + 4.99i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.999 + 0.0306i)5-s + 0.408·6-s + (−0.818 − 0.818i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.484 + 0.515i)10-s − 1.71·11-s + (0.204 − 0.204i)12-s + (−1.08 − 1.08i)13-s − 0.818·14-s + (−0.420 − 0.395i)15-s − 0.250·16-s + (1.21 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.982 + 0.185i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0589393 - 0.629036i\)
\(L(\frac12)\) \(\approx\) \(0.0589393 - 0.629036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.23 - 0.0685i)T \)
19 \( 1 + (4.01 + 1.68i)T \)
good7 \( 1 + (2.16 + 2.16i)T + 7iT^{2} \)
11 \( 1 + 5.68T + 11T^{2} \)
13 \( 1 + (3.92 + 3.92i)T + 13iT^{2} \)
17 \( 1 + (-4.99 - 4.99i)T + 17iT^{2} \)
23 \( 1 + (-4.30 + 4.30i)T - 23iT^{2} \)
29 \( 1 - 4.04T + 29T^{2} \)
31 \( 1 + 4.04iT - 31T^{2} \)
37 \( 1 + (6.19 - 6.19i)T - 37iT^{2} \)
41 \( 1 + 6.38iT - 41T^{2} \)
43 \( 1 + (4.15 - 4.15i)T - 43iT^{2} \)
47 \( 1 + (-3.24 - 3.24i)T + 47iT^{2} \)
53 \( 1 + (5.40 + 5.40i)T + 53iT^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + 2.33T + 61T^{2} \)
67 \( 1 + (-6.12 + 6.12i)T - 67iT^{2} \)
71 \( 1 + 6.51iT - 71T^{2} \)
73 \( 1 + (2.07 - 2.07i)T - 73iT^{2} \)
79 \( 1 + 4.23T + 79T^{2} \)
83 \( 1 + (5.30 - 5.30i)T - 83iT^{2} \)
89 \( 1 - 2.50T + 89T^{2} \)
97 \( 1 + (6.22 - 6.22i)T - 97iT^{2} \)
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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

−10.43114316629679848254691532576, −9.913762130587281117586217178783, −8.376570702790618626350084811136, −7.80161700027686157926110010179, −6.77288627344062042609396916359, −5.32948699001391761608243988570, −4.49731011161652748072305666198, −3.38138238290603491994752727696, −2.74489528006966811766673092827, −0.26957826551861490946452591371, 2.57735728379683710846267350506, 3.29713657680686095274856778993, 4.75299793529029865075252443782, 5.54251650142180490036132425514, 6.93489318938108627588505333277, 7.40616973646626664854050126597, 8.314911262973241802778118696151, 9.177245338765331798984728485676, 10.17626124376666171324699411186, 11.47276018847867510142235144584

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