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Properties

Label 2-570-57.56-c1-0-6
Degree $2$
Conductor $570$
Sign $-0.208 - 0.977i$
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 + (−0.209 + 1.71i)3-s + 4-s + i·5-s + (−0.209 + 1.71i)6-s + 0.264·7-s + 8-s + (−2.91 − 0.719i)9-s + i·10-s + 2i·11-s + (−0.209 + 1.71i)12-s + 5.43i·13-s + 0.264·14-s + (−1.71 − 0.209i)15-s + 16-s − 3.28i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.120 + 0.992i)3-s + 0.5·4-s + 0.447i·5-s + (−0.0854 + 0.701i)6-s + 0.0999·7-s + 0.353·8-s + (−0.970 − 0.239i)9-s + 0.316i·10-s + 0.603i·11-s + (−0.0603 + 0.496i)12-s + 1.50i·13-s + 0.0707·14-s + (−0.443 − 0.0540i)15-s + 0.250·16-s − 0.796i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.208 - 0.977i$
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: \((2,\ 570,\ (\ :1/2),\ -0.208 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26363 + 1.56192i\)
\(L(\frac12)\) \(\approx\) \(1.26363 + 1.56192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.209 - 1.71i)T \)
5 \( 1 - iT \)
19 \( 1 + (-1.41 - 4.12i)T \)
good7 \( 1 - 0.264T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 5.43iT - 13T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
23 \( 1 + 3.96iT - 23T^{2} \)
29 \( 1 + 0.418T + 29T^{2} \)
31 \( 1 - 3.56iT - 31T^{2} \)
37 \( 1 - 0.307iT - 37T^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 - 1.71T + 43T^{2} \)
47 \( 1 + 5.71iT - 47T^{2} \)
53 \( 1 + 3.29T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 9.18T + 61T^{2} \)
67 \( 1 + 0.666iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 8.08iT - 79T^{2} \)
83 \( 1 + 16.7iT - 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 1.47iT - 97T^{2} \)
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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

−11.11971235295374623603235633930, −10.13221100688497170255035777905, −9.500534542541188916996585790538, −8.421148287596227396509275978509, −7.13934470195501956072388516797, −6.35345354097103995529553580663, −5.20333742222568369559976169345, −4.39771403271867950298485686254, −3.49744891489799794145811560028, −2.20753811433048255698674252192, 0.964102993346722783525585099781, 2.50960639568917649793949192692, 3.63633754992180692354309965762, 5.19139004986046813834226714286, 5.75272034332472399502947654924, 6.75451156879360330192910329770, 7.84125443551053808024436419953, 8.340361069582477078758554599636, 9.623350895441413755700142614480, 10.93148750381594276792480548395

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