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Properties

Label 2-570-95.18-c1-0-9
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  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (1.89 + 1.18i)5-s + 1.00·6-s + (0.705 − 0.705i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (−0.498 − 2.17i)10-s + 1.32·11-s + (−0.707 − 0.707i)12-s + (0.741 − 0.741i)13-s − 0.997·14-s + (−2.17 + 0.498i)15-s − 1.00·16-s + (2.17 − 2.17i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (0.846 + 0.531i)5-s + 0.408·6-s + (0.266 − 0.266i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.157 − 0.689i)10-s + 0.400·11-s + (−0.204 − 0.204i)12-s + (0.205 − 0.205i)13-s − 0.266·14-s + (−0.562 + 0.128i)15-s − 0.250·16-s + (0.527 − 0.527i)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: $\chi_{570} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20672 + 0.145138i\)
\(L(\frac12)\) \(\approx\) \(1.20672 + 0.145138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.89 - 1.18i)T \)
19 \( 1 + (0.939 - 4.25i)T \)
good7 \( 1 + (-0.705 + 0.705i)T - 7iT^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + (-0.741 + 0.741i)T - 13iT^{2} \)
17 \( 1 + (-2.17 + 2.17i)T - 17iT^{2} \)
23 \( 1 + (1.08 + 1.08i)T + 23iT^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 - 5.95iT - 31T^{2} \)
37 \( 1 + (-1.33 - 1.33i)T + 37iT^{2} \)
41 \( 1 + 0.531iT - 41T^{2} \)
43 \( 1 + (-1.53 - 1.53i)T + 43iT^{2} \)
47 \( 1 + (-4.13 + 4.13i)T - 47iT^{2} \)
53 \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \)
59 \( 1 - 3.53T + 59T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 + (-1.99 - 1.99i)T + 67iT^{2} \)
71 \( 1 - 8.71iT - 71T^{2} \)
73 \( 1 + (5.83 + 5.83i)T + 73iT^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + (9.50 + 9.50i)T + 83iT^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (1.11 + 1.11i)T + 97iT^{2} \)
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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

−10.41081015805394350650443101349, −10.23932922877603197407667955820, −9.247676152808143703171998793609, −8.319386776078260277956109521874, −7.16120222093711734931729075843, −6.24358739538140913453855509065, −5.22587069186431380453719739303, −3.95387397978695162086052693943, −2.79235058106618836462274465350, −1.31806937097873496580790855478, 1.06443692025271496493354410452, 2.32003951494753541235983530695, 4.37804747427398763969446796852, 5.46711990384104723363427506646, 6.14510760261257280716986907895, 7.02147570529489081061475056325, 8.147408790241143083050414635512, 8.881819196960020879493769309258, 9.703554377550679530851685632073, 10.57255580101063629119879972813

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