Properties

Label 2-570-57.56-c1-0-12
Degree $2$
Conductor $570$
Sign $0.914 - 0.404i$
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  + 2-s + (−0.828 + 1.52i)3-s + 4-s i·5-s + (−0.828 + 1.52i)6-s + 4.83·7-s + 8-s + (−1.62 − 2.52i)9-s i·10-s − 2i·11-s + (−0.828 + 1.52i)12-s + 1.04i·13-s + 4.83·14-s + (1.52 + 0.828i)15-s + 16-s + 0.140i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.478 + 0.878i)3-s + 0.5·4-s − 0.447i·5-s + (−0.338 + 0.620i)6-s + 1.82·7-s + 0.353·8-s + (−0.542 − 0.840i)9-s − 0.316i·10-s − 0.603i·11-s + (−0.239 + 0.439i)12-s + 0.288i·13-s + 1.29·14-s + (0.392 + 0.213i)15-s + 0.250·16-s + 0.0339i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21001 + 0.466691i\)
\(L(\frac12)\) \(\approx\) \(2.21001 + 0.466691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.828 - 1.52i)T \)
5 \( 1 + iT \)
19 \( 1 + (-2.65 + 3.45i)T \)
good7 \( 1 - 4.83T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 1.04iT - 13T^{2} \)
17 \( 1 - 0.140iT - 17T^{2} \)
23 \( 1 - 6.63iT - 23T^{2} \)
29 \( 1 + 1.65T + 29T^{2} \)
31 \( 1 - 3.58iT - 31T^{2} \)
37 \( 1 - 6.36iT - 37T^{2} \)
41 \( 1 + 1.18T + 41T^{2} \)
43 \( 1 + 8.76T + 43T^{2} \)
47 \( 1 + 4.76iT - 47T^{2} \)
53 \( 1 - 8.42T + 53T^{2} \)
59 \( 1 - 0.350T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 2.83T + 73T^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 7.67iT - 97T^{2} \)
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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

−11.15648165677127295917132329510, −10.13241922869102860839000278490, −8.999743545250762201362075047290, −8.233428333138003747627912220837, −7.12326098786524890458460461463, −5.74085337634174894336338968406, −5.08520936425514045040351055959, −4.46373620669476606893037496390, −3.28907727686202117499889749047, −1.48543300797896228836544286738, 1.51477449168607220591880520593, 2.51734000833866675669940784817, 4.25752429350191604217897996941, 5.17623454458643915338880145215, 5.97321948829201935432346440706, 7.14824913347893468866812687173, 7.72625904813048555210817072646, 8.540979316603466810279749919367, 10.24701004944703738801201577232, 10.94938073169735128172872860810

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