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Properties

Label 2-570-95.12-c1-0-5
Degree $2$
Conductor $570$
Sign $0.704 - 0.710i$
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  + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (1.17 + 1.90i)5-s + (0.499 − 0.866i)6-s + (−2.04 − 2.04i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.643 − 2.14i)10-s + 5.53·11-s + (−0.707 + 0.707i)12-s + (2.58 − 0.692i)13-s + (1.44 + 2.49i)14-s + (−2.14 + 0.643i)15-s + (0.500 + 0.866i)16-s + (1.36 − 5.09i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (0.525 + 0.850i)5-s + (0.204 − 0.353i)6-s + (−0.771 − 0.771i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.203 − 0.677i)10-s + 1.66·11-s + (−0.204 + 0.204i)12-s + (0.716 − 0.191i)13-s + (0.385 + 0.668i)14-s + (−0.552 + 0.166i)15-s + (0.125 + 0.216i)16-s + (0.330 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.704 - 0.710i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.704 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04041 + 0.433455i\)
\(L(\frac12)\) \(\approx\) \(1.04041 + 0.433455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-1.17 - 1.90i)T \)
19 \( 1 + (-1.30 - 4.15i)T \)
good7 \( 1 + (2.04 + 2.04i)T + 7iT^{2} \)
11 \( 1 - 5.53T + 11T^{2} \)
13 \( 1 + (-2.58 + 0.692i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.36 + 5.09i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.42 - 5.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.57 + 4.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
37 \( 1 + (4.21 - 4.21i)T - 37iT^{2} \)
41 \( 1 + (7.73 - 4.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.27 + 1.41i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.87 + 1.84i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.55 + 1.75i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.47 - 7.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.35 + 5.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0830 - 0.309i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.99 + 5.76i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.4 - 2.81i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.715 - 1.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.42 - 2.42i)T - 83iT^{2} \)
89 \( 1 + (1.87 - 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.14 + 1.37i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.60139103720758857711474422827, −9.885588174948442337927684457165, −9.484283399030340973806599800901, −8.405170577073712304863670870301, −6.95476884084609272430136948444, −6.68668180932174051939203946104, −5.47439350768369677052008304367, −3.73794517092246961260790989245, −3.22093501832577053302758669221, −1.31590250981424993462840884555, 1.00038801142594060862206241814, 2.20376351329853495428113520271, 3.86990340546378156598991981791, 5.43013701855253185145783962007, 6.31659776902765152861135753562, 6.76704072594361793976524632519, 8.293946867516110522272552638260, 8.918895559752248786796691067383, 9.399753920899978424585451581341, 10.49870233238265415289042622247

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