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Properties

Label 2-570-285.83-c1-0-19
Degree $2$
Conductor $570$
Sign $0.934 - 0.355i$
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  + (−0.258 + 0.965i)2-s + (−0.158 + 1.72i)3-s + (−0.866 − 0.499i)4-s + (−1.67 − 1.48i)5-s + (−1.62 − 0.599i)6-s + (2.12 − 2.12i)7-s + (0.707 − 0.707i)8-s + (−2.94 − 0.548i)9-s + (1.86 − 1.23i)10-s − 0.171i·11-s + (0.999 − 1.41i)12-s + (2.73 − 0.732i)13-s + (1.5 + 2.59i)14-s + (2.82 − 2.64i)15-s + (0.500 + 0.866i)16-s + (0.580 − 2.16i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.0917 + 0.995i)3-s + (−0.433 − 0.249i)4-s + (−0.748 − 0.663i)5-s + (−0.663 − 0.244i)6-s + (0.801 − 0.801i)7-s + (0.249 − 0.249i)8-s + (−0.983 − 0.182i)9-s + (0.590 − 0.389i)10-s − 0.0517i·11-s + (0.288 − 0.408i)12-s + (0.757 − 0.203i)13-s + (0.400 + 0.694i)14-s + (0.729 − 0.684i)15-s + (0.125 + 0.216i)16-s + (0.140 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09960 + 0.202254i\)
\(L(\frac12)\) \(\approx\) \(1.09960 + 0.202254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.158 - 1.72i)T \)
5 \( 1 + (1.67 + 1.48i)T \)
19 \( 1 + (-4.33 + 0.5i)T \)
good7 \( 1 + (-2.12 + 2.12i)T - 7iT^{2} \)
11 \( 1 + 0.171iT - 11T^{2} \)
13 \( 1 + (-2.73 + 0.732i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.580 + 2.16i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.776 + 2.89i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.58 + 2.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (2.12 - 2.12i)T - 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 + 4.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.26 - 1.67i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.45 - 9.16i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 - 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.46 + 5.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.16 + 5.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.20 + 15.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.61 - 3.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 3i)T - 83iT^{2} \)
89 \( 1 + (-5.91 + 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 3.74i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.74359792966507181120922234823, −9.827171913099816512508309501987, −8.892157525228852574730033684695, −8.162454558147540871625443306851, −7.47415932060535398398477276605, −6.12298776721922936213662475798, −4.97086818201363925567468567722, −4.45730492142474983027025772248, −3.40159227801415237635013026074, −0.820289303098806223278579690597, 1.33936069611921193826091572537, 2.58512831988562123357800604425, 3.65736881544745113415374770343, 5.12146070129028367375360832843, 6.20709311427532907249911460389, 7.28810368905972326114211626515, 8.171045010826783481535321331121, 8.629478666603399524666039917149, 9.952245138520673774018830225648, 11.11721274962553327303461515614

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