Properties

Label 2-570-285.284-c1-0-20
Degree $2$
Conductor $570$
Sign $0.459 - 0.887i$
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 + (1.73 + 1.41i)5-s + (1 + 1.41i)6-s + 2.44i·7-s i·8-s + (1.00 − 2.82i)9-s + (−1.41 + 1.73i)10-s + 1.41i·11-s + (−1.41 + i)12-s + 4.24·13-s − 2.44·14-s + (3.86 + 0.267i)15-s + 16-s − 6.92·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.816 − 0.577i)3-s − 0.5·4-s + (0.774 + 0.632i)5-s + (0.408 + 0.577i)6-s + 0.925i·7-s − 0.353i·8-s + (0.333 − 0.942i)9-s + (−0.447 + 0.547i)10-s + 0.426i·11-s + (−0.408 + 0.288i)12-s + 1.17·13-s − 0.654·14-s + (0.997 + 0.0691i)15-s + 0.250·16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.459 - 0.887i$
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: \((2,\ 570,\ (\ :1/2),\ 0.459 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77046 + 1.07687i\)
\(L(\frac12)\) \(\approx\) \(1.77046 + 1.07687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 1.41i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 2.44T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−10.79841660906424039730909438086, −9.559471824343587157143558040211, −9.107730096612012138912099287640, −8.212272003128964222119770684347, −7.28817821247907017148069853043, −6.29924481020775683412890395392, −5.85608243995193203622258917694, −4.24772602455232353809365655064, −2.90165344209995032257333272853, −1.82701818561583760460637274292, 1.28120733755846271928916825873, 2.62233891280913395759714076567, 3.87033497310690517431515260208, 4.58317267789085601599603807991, 5.76574549789113542450504647263, 7.11018685262908510218373045466, 8.440272275862917848297377120413, 8.896493302555917597315301269739, 9.702943753807862942401007006033, 10.65171534505724186868597983305

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