Properties

Label 2-570-285.29-c1-0-21
Degree $2$
Conductor $570$
Sign $0.815 - 0.579i$
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  + (0.984 − 0.173i)2-s + (0.307 + 1.70i)3-s + (0.939 − 0.342i)4-s + (2.17 − 0.533i)5-s + (0.598 + 1.62i)6-s + (0.260 + 0.150i)7-s + (0.866 − 0.5i)8-s + (−2.81 + 1.04i)9-s + (2.04 − 0.902i)10-s + (1.17 − 0.680i)11-s + (0.871 + 1.49i)12-s + (2.32 + 1.94i)13-s + (0.282 + 0.102i)14-s + (1.57 + 3.53i)15-s + (0.766 − 0.642i)16-s + (−0.0110 − 0.0629i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.177 + 0.984i)3-s + (0.469 − 0.171i)4-s + (0.971 − 0.238i)5-s + (0.244 + 0.663i)6-s + (0.0985 + 0.0569i)7-s + (0.306 − 0.176i)8-s + (−0.936 + 0.349i)9-s + (0.646 − 0.285i)10-s + (0.355 − 0.205i)11-s + (0.251 + 0.432i)12-s + (0.644 + 0.540i)13-s + (0.0756 + 0.0275i)14-s + (0.407 + 0.913i)15-s + (0.191 − 0.160i)16-s + (−0.00269 − 0.0152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.815 - 0.579i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56777 + 0.819786i\)
\(L(\frac12)\) \(\approx\) \(2.56777 + 0.819786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.307 - 1.70i)T \)
5 \( 1 + (-2.17 + 0.533i)T \)
19 \( 1 + (3.34 + 2.79i)T \)
good7 \( 1 + (-0.260 - 0.150i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.17 + 0.680i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.32 - 1.94i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.0110 + 0.0629i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.311 - 0.113i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.30 - 7.39i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.24T + 37T^{2} \)
41 \( 1 + (4.00 - 3.35i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.86 + 5.13i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.167 + 0.949i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.00 - 5.49i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.12 - 6.40i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.91 + 1.06i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.99 + 11.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.92 + 2.88i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.48 - 7.72i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (9.29 + 11.0i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.598 - 1.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.2 + 8.62i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.21 - 6.90i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.81282771618496082567174350156, −10.07875416191113820572434612153, −9.062430182900208720227851419048, −8.589274495148315936059512927756, −6.92429596993858644842864511731, −5.98053715835330782631901965973, −5.15814678983584136644609997765, −4.25384752342657838363514288936, −3.16776299758057141854424775793, −1.86755706477461203508375016935, 1.54880202460225211151753434774, 2.60108158905042266404804691538, 3.84383000551365584698727421329, 5.38039273815845484688980584198, 6.14217902437143329389309014177, 6.78280775557002741645560281859, 7.85357674395622638083644349691, 8.700590372863239066985268255774, 9.838266904178724749658014431808, 10.82286717549100399738634032876

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