Properties

Label 2-570-15.8-c1-0-15
Degree $2$
Conductor $570$
Sign $0.637 - 0.770i$
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 + (1.34 + 1.08i)3-s + 1.00i·4-s + (−0.539 + 2.17i)5-s + (−0.182 − 1.72i)6-s + (2.86 − 2.86i)7-s + (0.707 − 0.707i)8-s + (0.629 + 2.93i)9-s + (1.91 − 1.15i)10-s + 4.60i·11-s + (−1.08 + 1.34i)12-s + (1.32 + 1.32i)13-s − 4.05·14-s + (−3.08 + 2.33i)15-s − 1.00·16-s + (−3.40 − 3.40i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.777 + 0.628i)3-s + 0.500i·4-s + (−0.241 + 0.970i)5-s + (−0.0746 − 0.703i)6-s + (1.08 − 1.08i)7-s + (0.250 − 0.250i)8-s + (0.209 + 0.977i)9-s + (0.605 − 0.364i)10-s + 1.38i·11-s + (−0.314 + 0.388i)12-s + (0.368 + 0.368i)13-s − 1.08·14-s + (−0.797 + 0.603i)15-s − 0.250·16-s + (−0.825 − 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38862 + 0.653253i\)
\(L(\frac12)\) \(\approx\) \(1.38862 + 0.653253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 - 1.08i)T \)
5 \( 1 + (0.539 - 2.17i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-2.86 + 2.86i)T - 7iT^{2} \)
11 \( 1 - 4.60iT - 11T^{2} \)
13 \( 1 + (-1.32 - 1.32i)T + 13iT^{2} \)
17 \( 1 + (3.40 + 3.40i)T + 17iT^{2} \)
23 \( 1 + (-2.00 + 2.00i)T - 23iT^{2} \)
29 \( 1 - 6.21T + 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 + (4.28 - 4.28i)T - 37iT^{2} \)
41 \( 1 - 1.74iT - 41T^{2} \)
43 \( 1 + (-8.19 - 8.19i)T + 43iT^{2} \)
47 \( 1 + (-5.22 - 5.22i)T + 47iT^{2} \)
53 \( 1 + (-4.96 + 4.96i)T - 53iT^{2} \)
59 \( 1 + 5.08T + 59T^{2} \)
61 \( 1 + 0.446T + 61T^{2} \)
67 \( 1 + (-6.27 + 6.27i)T - 67iT^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 + (6.36 + 6.36i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (3.87 - 3.87i)T - 83iT^{2} \)
89 \( 1 + 2.07T + 89T^{2} \)
97 \( 1 + (-12.5 + 12.5i)T - 97iT^{2} \)
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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

−10.81511981732370155148705729452, −10.07532648294609595898914653017, −9.285226766528321381165583047623, −8.209668584809181837348833473521, −7.46127002712289057949770703408, −6.84319033909030053735075269369, −4.66766058171065963103850285252, −4.20807504700735195458845851022, −2.91324579962580112076153368207, −1.79763109741853381135407145031, 1.04580901559119128591958372138, 2.30183323521474437963241378308, 3.90956486587833298596582367443, 5.33718872390153364985503597559, 5.99692963791981390661497035475, 7.33412795737439498169500623961, 8.283311215719968182239408979001, 8.772100439691316268898785126786, 8.975074152511505774829094585377, 10.62358769762886874586898248352

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