Properties

Label 2-570-5.4-c1-0-11
Degree $2$
Conductor $570$
Sign $0.447 + 0.894i$
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  + i·2-s + i·3-s − 4-s + (−1 − 2i)5-s − 6-s i·8-s − 9-s + (2 − i)10-s − 4·11-s i·12-s − 2i·13-s + (2 − i)15-s + 16-s − 6i·17-s i·18-s + 19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s + (0.516 − 0.258i)15-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571051 - 0.352928i\)
\(L(\frac12)\) \(\approx\) \(0.571051 - 0.352928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (1 + 2i)T \)
19 \( 1 - T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.46462821165226532758204302613, −9.517752060107356672020385457221, −8.766363184929676765386913323261, −7.930333423499553177034582419579, −7.20021440091719279360429174353, −5.68731223812146267683738564969, −5.07455768381776820349955560035, −4.22696197960079737462113218221, −2.83488330187070679192426662326, −0.37345075326660318719524411743, 1.79229596142358938527930914928, 2.96184052688672467759281624440, 3.93427986242902300787553498051, 5.34485935136836988743198872688, 6.40534708988336544380642055374, 7.49629294753819375148836117429, 8.106015319848450357798688585191, 9.210128340046148514239332393854, 10.42780428767633370412130043710, 10.76378627726636236112099785040

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