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Properties

Label 2-570-57.50-c1-0-15
Degree $2$
Conductor $570$
Sign $0.971 + 0.237i$
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  + (−0.5 − 0.866i)2-s + (1.37 + 1.05i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (0.224 − 1.71i)6-s − 1.74·7-s + 0.999·8-s + (0.780 + 2.89i)9-s + (−0.866 − 0.499i)10-s − 4.48i·11-s + (−1.59 + 0.663i)12-s + (3.14 + 1.81i)13-s + (0.871 + 1.51i)14-s + (1.71 + 0.224i)15-s + (−0.5 − 0.866i)16-s + (5.72 − 3.30i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.793 + 0.608i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.0917 − 0.701i)6-s − 0.659·7-s + 0.353·8-s + (0.260 + 0.965i)9-s + (−0.273 − 0.158i)10-s − 1.35i·11-s + (−0.461 + 0.191i)12-s + (0.872 + 0.503i)13-s + (0.232 + 0.403i)14-s + (0.443 + 0.0580i)15-s + (−0.125 − 0.216i)16-s + (1.38 − 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65544 - 0.199439i\)
\(L(\frac12)\) \(\approx\) \(1.65544 - 0.199439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.37 - 1.05i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-3.74 - 2.22i)T \)
good7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 + 4.48iT - 11T^{2} \)
13 \( 1 + (-3.14 - 1.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.72 + 3.30i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.69 - 0.977i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.271 - 0.469i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + 0.251iT - 37T^{2} \)
41 \( 1 + (-2.28 - 3.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.13 + 3.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.28 - 0.743i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0649 - 0.112i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.22 + 5.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.40 + 4.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.78 + 8.29i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.51 + 6.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.04 - 2.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (2.10 - 3.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.01 + 5.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.52125714647738991493289794604, −9.751418776081630127443611355193, −9.089457960052512458457910065001, −8.416621690929310844204018331393, −7.42959239654982312054599199351, −6.01945581800616964784246364351, −4.96849972381003843005042906771, −3.42122723079917781561320519616, −3.15240487022106584360707685936, −1.34693849435748862282454699823, 1.31985031991689615234838928127, 2.78431180828351611596532750642, 3.98123609600267292544259654155, 5.58210099963160970900360073628, 6.41683904863122395476478586035, 7.32959685983587309611043289362, 7.927494788049267900212156865865, 9.013007817541225912425533255748, 9.743729731694036827959181158584, 10.31488185909547281717572395669

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