Properties

Label 2-570-15.8-c1-0-13
Degree $2$
Conductor $570$
Sign $0.331 - 0.943i$
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 + (−0.581 − 1.63i)3-s + 1.00i·4-s + (1.36 + 1.76i)5-s + (0.742 − 1.56i)6-s + (0.102 − 0.102i)7-s + (−0.707 + 0.707i)8-s + (−2.32 + 1.89i)9-s + (−0.282 + 2.21i)10-s + 6.12i·11-s + (1.63 − 0.581i)12-s + (−0.754 − 0.754i)13-s + 0.145·14-s + (2.08 − 3.26i)15-s − 1.00·16-s + (2.33 + 2.33i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.335 − 0.941i)3-s + 0.500i·4-s + (0.612 + 0.790i)5-s + (0.302 − 0.638i)6-s + (0.0388 − 0.0388i)7-s + (−0.250 + 0.250i)8-s + (−0.774 + 0.632i)9-s + (−0.0893 + 0.701i)10-s + 1.84i·11-s + (0.470 − 0.167i)12-s + (−0.209 − 0.209i)13-s + 0.0388·14-s + (0.539 − 0.842i)15-s − 0.250·16-s + (0.565 + 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.331 - 0.943i$
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: \((2,\ 570,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42786 + 1.01198i\)
\(L(\frac12)\) \(\approx\) \(1.42786 + 1.01198i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.581 + 1.63i)T \)
5 \( 1 + (-1.36 - 1.76i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.102 + 0.102i)T - 7iT^{2} \)
11 \( 1 - 6.12iT - 11T^{2} \)
13 \( 1 + (0.754 + 0.754i)T + 13iT^{2} \)
17 \( 1 + (-2.33 - 2.33i)T + 17iT^{2} \)
23 \( 1 + (-6.21 + 6.21i)T - 23iT^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (3.66 - 3.66i)T - 37iT^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (0.994 + 0.994i)T + 43iT^{2} \)
47 \( 1 + (-4.88 - 4.88i)T + 47iT^{2} \)
53 \( 1 + (2.86 - 2.86i)T - 53iT^{2} \)
59 \( 1 - 7.12T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + (-4.37 + 4.37i)T - 67iT^{2} \)
71 \( 1 - 0.311iT - 71T^{2} \)
73 \( 1 + (-0.755 - 0.755i)T + 73iT^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 + 2.30T + 89T^{2} \)
97 \( 1 + (0.595 - 0.595i)T - 97iT^{2} \)
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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

−10.88073645700074864663028111574, −10.22638496137516407613067672074, −9.059489282193841346763485652694, −7.79786169090768890493140248064, −7.13569671340677378842761389096, −6.51653741536174390204458773202, −5.55979568385537762606746395864, −4.57855447129265411315535922495, −2.92445914629581224653929655827, −1.85573165322041027577092360839, 0.936245169091678200732961637435, 2.89015298111108977465857108119, 3.83229470529222203912499388600, 5.18237407435322550416174506469, 5.44517119865888835670182111096, 6.53355941265798988474028891667, 8.251059538883220634074575409277, 9.150385279465683601903817503559, 9.667494866274425806950141458691, 10.71095019118114029936488688392

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