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Properties

Label 2-570-19.7-c1-0-8
Degree $2$
Conductor $570$
Sign $0.469 + 0.882i$
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 + 3.84·7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)10-s + 5.73·11-s + 0.999·12-s + (−2.36 + 4.10i)13-s + (−1.92 − 3.32i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (3.31 + 5.73i)17-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 + 1.45·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + 1.72·11-s + 0.288·12-s + (−0.656 + 1.13i)13-s + (−0.513 − 0.889i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.803 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.469 + 0.882i$
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: \((2,\ 570,\ (\ :1/2),\ 0.469 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11389 - 0.669313i\)
\(L(\frac12)\) \(\approx\) \(1.11389 - 0.669313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 - 0.771i)T \)
good7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 - 5.73T + 11T^{2} \)
13 \( 1 + (2.36 - 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.31 - 5.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.86 + 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.15 + 8.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (4.39 + 7.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.523 + 0.905i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.890 - 1.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.55 - 4.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0545 + 0.0945i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.94 - 5.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.31 - 5.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.05 - 5.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.36 - 5.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-2.55 + 4.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.57 + 7.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77018598430280860822896954522, −9.767149826422005748342813391975, −8.542951801469666817869137996614, −8.332695267823195846665438816530, −7.06699139820154703123095522351, −6.13581524723947424067145108365, −4.60957886250076402470296510932, −4.09544920706208513110676909484, −2.08356343328214515761324607637, −1.22497020132138290534711062934, 1.22474612823224894079313070575, 3.21806806561865277888249073568, 4.65890364921107363387361536416, 5.18898059335537465772979616760, 6.47802388095182162533311066905, 7.33723734877583889970134751911, 8.211232446238042124909326014334, 9.080924214240435201836042817692, 9.962062114411944769444365067679, 10.85190494196035653679382784675

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