Properties

Label 2-570-19.7-c1-0-10
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: $\chi_{570} (121, \cdot )$
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

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

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