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Properties

Label 2-570-15.8-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.997 - 0.0657i$
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 + (−1.66 + 0.492i)3-s + 1.00i·4-s + (1.86 + 1.23i)5-s + (−1.52 − 0.825i)6-s + (−3.33 + 3.33i)7-s + (−0.707 + 0.707i)8-s + (2.51 − 1.63i)9-s + (0.442 + 2.19i)10-s − 0.00329i·11-s + (−0.492 − 1.66i)12-s + (1.14 + 1.14i)13-s − 4.71·14-s + (−3.70 − 1.13i)15-s − 1.00·16-s + (−4.95 − 4.95i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.958 + 0.284i)3-s + 0.500i·4-s + (0.833 + 0.553i)5-s + (−0.621 − 0.337i)6-s + (−1.26 + 1.26i)7-s + (−0.250 + 0.250i)8-s + (0.838 − 0.545i)9-s + (0.139 + 0.693i)10-s − 0.000994i·11-s + (−0.142 − 0.479i)12-s + (0.318 + 0.318i)13-s − 1.26·14-s + (−0.956 − 0.293i)15-s − 0.250·16-s + (−1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.997 - 0.0657i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.997 - 0.0657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0335011 + 1.01726i\)
\(L(\frac12)\) \(\approx\) \(0.0335011 + 1.01726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 0.492i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
19 \( 1 - iT \)
good7 \( 1 + (3.33 - 3.33i)T - 7iT^{2} \)
11 \( 1 + 0.00329iT - 11T^{2} \)
13 \( 1 + (-1.14 - 1.14i)T + 13iT^{2} \)
17 \( 1 + (4.95 + 4.95i)T + 17iT^{2} \)
23 \( 1 + (1.28 - 1.28i)T - 23iT^{2} \)
29 \( 1 - 1.91T + 29T^{2} \)
31 \( 1 + 3.49T + 31T^{2} \)
37 \( 1 + (6.71 - 6.71i)T - 37iT^{2} \)
41 \( 1 - 3.63iT - 41T^{2} \)
43 \( 1 + (-1.52 - 1.52i)T + 43iT^{2} \)
47 \( 1 + (2.07 + 2.07i)T + 47iT^{2} \)
53 \( 1 + (-5.97 + 5.97i)T - 53iT^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 1.95T + 61T^{2} \)
67 \( 1 + (5.56 - 5.56i)T - 67iT^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 + (-10.3 - 10.3i)T + 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (-9.29 + 9.29i)T - 83iT^{2} \)
89 \( 1 + 6.59T + 89T^{2} \)
97 \( 1 + (-7.72 + 7.72i)T - 97iT^{2} \)
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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

−11.38846719243295943162017945682, −10.18885735805102574728093139741, −9.489626495842438115141704266306, −8.755682789978961292206895329892, −6.92629422837700065854218592419, −6.56091003485156808364791481668, −5.73538665032153681850986374258, −5.02314702478334885501130812390, −3.55209476834965610605940568272, −2.37921211161760915801928554599, 0.53003742285023130351779906069, 1.96059768695099538659483878162, 3.71087422000719690708267010572, 4.59488955864292812701395818632, 5.79595841555664150560016663174, 6.40372105572771665377625750761, 7.21762378144715786164939251224, 8.780766487984251923689724478838, 9.808703549420903566020030490878, 10.56928027077264478797297984320

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