Properties

Label 2-570-285.284-c1-0-21
Degree $2$
Conductor $570$
Sign $0.761 - 0.648i$
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  + i·2-s + (−1.41 + i)3-s − 4-s + (2.12 − 0.707i)5-s + (−1 − 1.41i)6-s i·8-s + (1.00 − 2.82i)9-s + (0.707 + 2.12i)10-s − 2.82i·11-s + (1.41 − i)12-s + 4.24·13-s + (−2.29 + 3.12i)15-s + 16-s + (2.82 + 1.00i)18-s + (1 − 4.24i)19-s + (−2.12 + 0.707i)20-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.816 + 0.577i)3-s − 0.5·4-s + (0.948 − 0.316i)5-s + (−0.408 − 0.577i)6-s − 0.353i·8-s + (0.333 − 0.942i)9-s + (0.223 + 0.670i)10-s − 0.852i·11-s + (0.408 − 0.288i)12-s + 1.17·13-s + (−0.592 + 0.805i)15-s + 0.250·16-s + (0.666 + 0.235i)18-s + (0.229 − 0.973i)19-s + (−0.474 + 0.158i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22347 + 0.450594i\)
\(L(\frac12)\) \(\approx\) \(1.22347 + 0.450594i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 + (-2.12 + 0.707i)T \)
19 \( 1 + (-1 + 4.24i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.96056770875468781036983388291, −9.770374555632791520173012973137, −9.154290147816431990271151020636, −8.362164639953322420386460978276, −6.89257716174919750285198260916, −6.13155290919767784587210879393, −5.46890835657423083265619613412, −4.58663052341913128737556087840, −3.27390827507081072546679474255, −1.03025907179804271530537988598, 1.31400295608587163549477759035, 2.31354768362619711898124154081, 3.88154751266298920017493264419, 5.22646297852615076052347844004, 5.94970730823940545140633868054, 6.90178140496929015553817338434, 7.925726476071281840270006666344, 9.172839970909093056028580540101, 9.973402328484605023970817469542, 10.78637054192516808623455130334

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