Properties

Label 2-570-57.56-c1-0-18
Degree $2$
Conductor $570$
Sign $0.236 + 0.971i$
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  − 2-s + (1.67 − 0.423i)3-s + 4-s i·5-s + (−1.67 + 0.423i)6-s − 2.47·7-s − 8-s + (2.64 − 1.42i)9-s + i·10-s − 2i·11-s + (1.67 − 0.423i)12-s + 1.15i·13-s + 2.47·14-s + (−0.423 − 1.67i)15-s + 16-s − 6.67i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.969 − 0.244i)3-s + 0.5·4-s − 0.447i·5-s + (−0.685 + 0.173i)6-s − 0.934·7-s − 0.353·8-s + (0.880 − 0.474i)9-s + 0.316i·10-s − 0.603i·11-s + (0.484 − 0.122i)12-s + 0.319i·13-s + 0.660·14-s + (−0.109 − 0.433i)15-s + 0.250·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.236 + 0.971i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01658 - 0.799147i\)
\(L(\frac12)\) \(\approx\) \(1.01658 - 0.799147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.67 + 0.423i)T \)
5 \( 1 + iT \)
19 \( 1 + (-4.35 - 0.0388i)T \)
good7 \( 1 + 2.47T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 1.15iT - 13T^{2} \)
17 \( 1 + 6.67iT - 17T^{2} \)
23 \( 1 + 5.79iT - 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 + 4.80iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 7.83T + 41T^{2} \)
43 \( 1 + 0.978T + 43T^{2} \)
47 \( 1 - 3.02iT - 47T^{2} \)
53 \( 1 + 2.33T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 + 7.24T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 5.75iT - 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 6.94iT - 97T^{2} \)
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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

−10.03846031916087198525421902819, −9.645659916593142150890156168612, −8.794553809677792666895670993225, −8.117132046056452857661158306095, −7.07229827817652047091864808636, −6.39014469209090615738762997765, −4.88275114047376043511820139011, −3.40745057945928375366645701001, −2.56196079785262663743072419009, −0.856071556048657012534188796177, 1.77745742364231431421247865690, 3.08358704447890703485588413940, 3.84585071863623973304417637436, 5.52830050211903046114216534010, 6.77647872772529472599620100914, 7.44586988872774273092883006887, 8.388294725079460078929974321868, 9.214557706365913457552199240776, 10.09914619002337600874600486332, 10.36148514433256066588034717825

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