Properties

Label 2-570-19.6-c1-0-4
Degree $2$
Conductor $570$
Sign $0.765 + 0.643i$
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  + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (0.705 − 1.22i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (2.76 + 4.79i)11-s + (−0.499 + 0.866i)12-s + (−4.69 + 1.71i)13-s + (0.245 + 1.39i)14-s + (−0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (5.94 − 4.98i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (0.383 − 0.139i)6-s + (0.266 − 0.462i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (0.833 + 1.44i)11-s + (−0.144 + 0.250i)12-s + (−1.30 + 0.474i)13-s + (0.0655 + 0.371i)14-s + (−0.0448 + 0.254i)15-s + (−0.234 − 0.0855i)16-s + (1.44 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833052 - 0.303514i\)
\(L(\frac12)\) \(\approx\) \(0.833052 - 0.303514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (2.23 + 3.74i)T \)
good7 \( 1 + (-0.705 + 1.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.76 - 4.79i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.69 - 1.71i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-5.94 + 4.98i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.44 + 8.20i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.34 - 1.13i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.71 + 4.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.50T + 37T^{2} \)
41 \( 1 + (-4.89 - 1.78i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.837 + 4.74i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.04 - 1.71i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.52 + 8.66i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (5.24 - 4.39i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.67 + 9.49i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.29 + 4.43i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.361 + 2.05i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (0.800 + 0.291i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.06 + 3.29i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.18 - 2.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-13.3 + 4.86i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-11.1 + 9.31i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.41304681910638229006280878318, −9.711619782581365598209843985478, −9.023942876544435882567517625244, −7.65944383323075917910678140619, −7.20449953901817950778736849845, −6.33558874803944169939883522172, −4.85352919546511728205953365283, −4.54695525907449977564774229488, −2.29449859182110137903216335871, −0.75013417158504110099771127527, 1.30155696186020984279144278043, 2.99528263254116291022937265793, 3.92776683667664695126528704876, 5.49493137974040945958901498448, 6.14120762502998551169434007876, 7.47138661293630882479568128030, 8.219193473987804738330140544951, 9.225985267337741142389947675665, 10.15139336178511050471966040558, 10.71010753085958839280257964083

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