Properties

Label 2-570-95.37-c1-0-7
Degree $2$
Conductor $570$
Sign $0.885 - 0.464i$
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (1.25 + 1.84i)5-s + 1.00·6-s + (3.10 + 3.10i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (2.19 + 0.416i)10-s − 3.82·11-s + (0.707 − 0.707i)12-s + (−0.0891 − 0.0891i)13-s + 4.39·14-s + (−0.416 + 2.19i)15-s − 1.00·16-s + (−1.83 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.562 + 0.826i)5-s + 0.408·6-s + (1.17 + 1.17i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.694 + 0.131i)10-s − 1.15·11-s + (0.204 − 0.204i)12-s + (−0.0247 − 0.0247i)13-s + 1.17·14-s + (−0.107 + 0.567i)15-s − 0.250·16-s + (−0.443 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.885 - 0.464i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34939 + 0.578370i\)
\(L(\frac12)\) \(\approx\) \(2.34939 + 0.578370i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.84i)T \)
19 \( 1 + (2.70 - 3.41i)T \)
good7 \( 1 + (-3.10 - 3.10i)T + 7iT^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + (0.0891 + 0.0891i)T + 13iT^{2} \)
17 \( 1 + (1.83 + 1.83i)T + 17iT^{2} \)
23 \( 1 + (-2.58 + 2.58i)T - 23iT^{2} \)
29 \( 1 - 3.60T + 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 + (-7.93 + 7.93i)T - 43iT^{2} \)
47 \( 1 + (0.463 + 0.463i)T + 47iT^{2} \)
53 \( 1 + (3.21 + 3.21i)T + 53iT^{2} \)
59 \( 1 - 9.40T + 59T^{2} \)
61 \( 1 - 8.21T + 61T^{2} \)
67 \( 1 + (8.78 - 8.78i)T - 67iT^{2} \)
71 \( 1 + 1.66iT - 71T^{2} \)
73 \( 1 + (3.64 - 3.64i)T - 73iT^{2} \)
79 \( 1 + 8.82T + 79T^{2} \)
83 \( 1 + (0.347 - 0.347i)T - 83iT^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + (6.88 - 6.88i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80182960764643501092745569303, −10.19211545553938763126387693663, −9.114493495731299659357880622285, −8.331723803323326287313386926755, −7.24943215112771226526024692621, −5.82104014711753653337360457234, −5.27841716320479361803577802123, −4.12317102176281662506809340798, −2.56815719386129822675360563184, −2.23975493599900291725633329393, 1.27964757098263930999947532782, 2.72355177145819187246500091094, 4.44514132205564527190089356059, 4.84367373310643424576213522059, 6.09928613070243074330344914493, 7.16720950322782517375921437433, 8.041905915427695811627202681543, 8.497689705732600894231902222352, 9.721108892328275234508266684609, 10.76348231394512554651625929772

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