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Properties

Label 2-570-95.8-c1-0-13
Degree $2$
Conductor $570$
Sign $0.787 + 0.615i$
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 + (2.01 + 0.977i)5-s + (−0.499 − 0.866i)6-s + (−0.109 + 0.109i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (2.19 + 0.424i)10-s + 2.55·11-s + (−0.707 − 0.707i)12-s + (0.833 + 0.223i)13-s + (−0.0772 + 0.133i)14-s + (0.424 − 2.19i)15-s + (0.500 − 0.866i)16-s + (−1.60 − 6.00i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.899 + 0.437i)5-s + (−0.204 − 0.353i)6-s + (−0.0413 + 0.0413i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.694 + 0.134i)10-s + 0.769·11-s + (−0.204 − 0.204i)12-s + (0.231 + 0.0619i)13-s + (−0.0206 + 0.0357i)14-s + (0.109 − 0.566i)15-s + (0.125 − 0.216i)16-s + (−0.390 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.787 + 0.615i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.787 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31852 - 0.798812i\)
\(L(\frac12)\) \(\approx\) \(2.31852 - 0.798812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 - 0.977i)T \)
19 \( 1 + (-4.19 - 1.19i)T \)
good7 \( 1 + (0.109 - 0.109i)T - 7iT^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 + (-0.833 - 0.223i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.60 + 6.00i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (2.33 - 8.70i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.90iT - 31T^{2} \)
37 \( 1 + (8.27 + 8.27i)T + 37iT^{2} \)
41 \( 1 + (1.06 + 0.615i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.66 - 1.24i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-9.57 - 2.56i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.213 - 0.0571i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.26 - 5.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.81 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 - 7.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.41 - 4.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.5 - 2.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.74 - 6.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.121 + 0.121i)T + 83iT^{2} \)
89 \( 1 + (-6.05 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.3 - 2.76i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.92720709969342585867354779462, −9.679009184311077014607385728942, −9.228258365651935120795472468217, −7.58093701188960277848650194103, −6.95157071392313250951070477179, −5.91261593987555190462318065463, −5.34273444444281660160932765955, −3.83871412992871887046108596267, −2.65584092243721269325954959336, −1.48210998467065186365360842954, 1.68020053444452756953675464325, 3.23459965851060001070811730682, 4.35707664594194489400864525570, 5.20967538387831222762917987058, 6.17949076216709122203257533387, 6.81844918778255966398502144631, 8.445530874030122965744236440418, 8.952738890691737351357964928032, 10.21747536258166681282781755368, 10.62005727327799734638590871294

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