Properties

Label 2-570-285.269-c1-0-32
Degree $2$
Conductor $570$
Sign $-0.502 + 0.864i$
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  + (0.642 − 0.766i)2-s + (0.110 − 1.72i)3-s + (−0.173 − 0.984i)4-s + (2.23 − 0.138i)5-s + (−1.25 − 1.19i)6-s + (2.23 − 1.28i)7-s + (−0.866 − 0.500i)8-s + (−2.97 − 0.381i)9-s + (1.32 − 1.79i)10-s + (−2.63 − 1.51i)11-s + (−1.72 + 0.191i)12-s + (3.00 + 1.09i)13-s + (0.447 − 2.53i)14-s + (0.00666 − 3.87i)15-s + (−0.939 + 0.342i)16-s + (1.98 + 1.66i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.0637 − 0.997i)3-s + (−0.0868 − 0.492i)4-s + (0.998 − 0.0620i)5-s + (−0.511 − 0.488i)6-s + (0.843 − 0.487i)7-s + (−0.306 − 0.176i)8-s + (−0.991 − 0.127i)9-s + (0.420 − 0.568i)10-s + (−0.793 − 0.458i)11-s + (−0.496 + 0.0552i)12-s + (0.832 + 0.302i)13-s + (0.119 − 0.678i)14-s + (0.00171 − 0.999i)15-s + (−0.234 + 0.0855i)16-s + (0.480 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11139 - 1.93199i\)
\(L(\frac12)\) \(\approx\) \(1.11139 - 1.93199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.110 + 1.72i)T \)
5 \( 1 + (-2.23 + 0.138i)T \)
19 \( 1 + (0.765 - 4.29i)T \)
good7 \( 1 + (-2.23 + 1.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.63 + 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.00 - 1.09i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.98 - 1.66i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.878 + 4.98i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.575 + 0.482i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (7.47 - 4.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.30T + 37T^{2} \)
41 \( 1 + (-0.243 + 0.0887i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (5.12 + 0.903i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.62 - 5.55i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (4.33 - 0.764i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-3.10 - 2.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.76 + 10.0i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.42 + 7.91i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.792 + 4.49i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.36 - 9.24i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-5.67 - 15.5i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.188 + 0.326i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.2 - 4.47i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (1.74 + 1.46i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.81404846204061979309501873991, −9.722353393217579117685364706464, −8.516109350286706464358255206487, −7.918721715518302533292917888577, −6.56984244433976488159492494646, −5.86216274705314076218357508513, −4.94984280617712039481980828641, −3.45814823419571937975976611533, −2.14327291087343727863965577660, −1.24264940285140345261577538170, 2.21998197369248804461085280169, 3.41715176745739516821961154417, 4.83676295427908062120104068563, 5.33232729828296285305103947066, 6.13293250295992831333272927337, 7.51659837331976245846900079114, 8.457697687317705859497057572597, 9.271099108099100562516083608579, 10.05807733897510995242021246308, 11.04720615504776206584705020589

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