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Properties

Label 2-570-15.8-c1-0-8
Degree $2$
Conductor $570$
Sign $-0.995 - 0.0968i$
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 + (−0.162 + 1.72i)3-s + 1.00i·4-s + (−0.520 + 2.17i)5-s + (−1.33 + 1.10i)6-s + (0.496 − 0.496i)7-s + (−0.707 + 0.707i)8-s + (−2.94 − 0.561i)9-s + (−1.90 + 1.17i)10-s + 1.60i·11-s + (−1.72 − 0.162i)12-s + (0.465 + 0.465i)13-s + 0.702·14-s + (−3.66 − 1.25i)15-s − 1.00·16-s + (0.134 + 0.134i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.0939 + 0.995i)3-s + 0.500i·4-s + (−0.232 + 0.972i)5-s + (−0.544 + 0.450i)6-s + (0.187 − 0.187i)7-s + (−0.250 + 0.250i)8-s + (−0.982 − 0.187i)9-s + (−0.602 + 0.370i)10-s + 0.482i·11-s + (−0.497 − 0.0469i)12-s + (0.128 + 0.128i)13-s + 0.187·14-s + (−0.946 − 0.322i)15-s − 0.250·16-s + (0.0326 + 0.0326i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.995 - 0.0968i$
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: \((2,\ 570,\ (\ :1/2),\ -0.995 - 0.0968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0735741 + 1.51628i\)
\(L(\frac12)\) \(\approx\) \(0.0735741 + 1.51628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.162 - 1.72i)T \)
5 \( 1 + (0.520 - 2.17i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.496 + 0.496i)T - 7iT^{2} \)
11 \( 1 - 1.60iT - 11T^{2} \)
13 \( 1 + (-0.465 - 0.465i)T + 13iT^{2} \)
17 \( 1 + (-0.134 - 0.134i)T + 17iT^{2} \)
23 \( 1 + (0.307 - 0.307i)T - 23iT^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 + (-4.03 + 4.03i)T - 37iT^{2} \)
41 \( 1 - 3.54iT - 41T^{2} \)
43 \( 1 + (-3.15 - 3.15i)T + 43iT^{2} \)
47 \( 1 + (4.45 + 4.45i)T + 47iT^{2} \)
53 \( 1 + (2.36 - 2.36i)T - 53iT^{2} \)
59 \( 1 - 8.58T + 59T^{2} \)
61 \( 1 - 6.10T + 61T^{2} \)
67 \( 1 + (6.28 - 6.28i)T - 67iT^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + (-2.24 - 2.24i)T + 73iT^{2} \)
79 \( 1 - 3.48iT - 79T^{2} \)
83 \( 1 + (-1.76 + 1.76i)T - 83iT^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (0.0173 - 0.0173i)T - 97iT^{2} \)
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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

−11.17347901735450196411217682868, −10.32019047956176978835203177704, −9.550588133865868346367556603399, −8.413866538350851337703872208733, −7.50659707014515904406937398158, −6.53492745019296829130127282212, −5.63614459624523235354178973248, −4.50510148149651035976960171958, −3.73326556676062427782797869066, −2.62496700401671085889086051381, 0.75129128341386053329252230567, 2.03884186260625424163844181981, 3.39380761132725163186314031766, 4.75279419149130025855575700837, 5.59849683668080403170000746642, 6.51202316275547076118238223167, 7.73470653786711417992940042761, 8.507180993144085693788620016602, 9.308907689895683455577216226487, 10.59080725472849516981343309874

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