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Properties

Label 2-570-19.11-c1-0-11
Degree $2$
Conductor $570$
Sign $-0.181 + 0.983i$
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  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.499 − 0.866i)6-s + 2.64·7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s + 4.29·11-s − 0.999·12-s + (1.32 − 2.29i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.177 − 0.306i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.204 − 0.353i)6-s + 0.999·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + 1.29·11-s − 0.288·12-s + (0.353 − 0.612i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.0429 − 0.0744i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.181 + 0.983i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37838 - 1.65536i\)
\(L(\frac12)\) \(\approx\) \(1.37838 - 1.65536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (4.32 - 0.559i)T \)
good7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.177 + 0.306i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.822 + 1.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 + 2.29T + 37T^{2} \)
41 \( 1 + (-1.67 + 2.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.29 - 7.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.322 + 0.559i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.64 + 4.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.46 - 7.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.17 - 3.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.17 + 7.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.17 - 7.23i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.64 + 4.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 + (-6.61 - 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.82 - 3.15i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70676035877132514622471518942, −9.541727049453656983672766136941, −8.802999207658375004371114095927, −8.005595976211235491246942205004, −6.79235315383597079819646683844, −5.82907693754770665699820378681, −4.67111069455917670056302021452, −3.78739608329031319221942992866, −2.22423637587724817772229916410, −1.26267219733485229774099777472, 1.94926380394777601894782614590, 3.54789851137010996776476931703, 4.39963802203713470232055659529, 5.40508462689383278049719813251, 6.45611960966774423908519883295, 7.33300887260345834928555608050, 8.416413982679193364886911322734, 9.004826625739133332128913227430, 10.05289678277111231650508964613, 11.04329176211298300166061217583

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