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Properties

Label 2-570-95.88-c1-0-10
Degree $2$
Conductor $570$
Sign $-0.0737 - 0.997i$
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 + (0.965 + 0.258i)3-s + (−0.866 − 0.499i)4-s + (0.599 + 2.15i)5-s + (−0.499 + 0.866i)6-s + (0.819 − 0.819i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−2.23 + 0.0214i)10-s + 2.57·11-s + (−0.707 − 0.707i)12-s + (−0.0277 − 0.103i)13-s + (0.579 + 1.00i)14-s + (0.0214 + 2.23i)15-s + (0.500 + 0.866i)16-s + (5.87 + 1.57i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.268 + 0.963i)5-s + (−0.204 + 0.353i)6-s + (0.309 − 0.309i)7-s + (0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.707 + 0.00677i)10-s + 0.775·11-s + (−0.204 − 0.204i)12-s + (−0.00770 − 0.0287i)13-s + (0.154 + 0.268i)14-s + (0.00553 + 0.577i)15-s + (0.125 + 0.216i)16-s + (1.42 + 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0737 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0737 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0737 - 0.997i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.0737 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16054 + 1.24958i\)
\(L(\frac12)\) \(\approx\) \(1.16054 + 1.24958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-0.599 - 2.15i)T \)
19 \( 1 + (2.43 - 3.61i)T \)
good7 \( 1 + (-0.819 + 0.819i)T - 7iT^{2} \)
11 \( 1 - 2.57T + 11T^{2} \)
13 \( 1 + (0.0277 + 0.103i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.87 - 1.57i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (4.66 - 1.25i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.920 + 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.46iT - 31T^{2} \)
37 \( 1 + (2.36 + 2.36i)T + 37iT^{2} \)
41 \( 1 + (-3.53 + 2.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 - 6.26i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.677 - 2.52i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.260 - 0.972i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.34 + 4.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.67 + 9.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.15 - 1.91i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.46 + 3.15i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.38 + 5.16i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.79 + 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.39 + 1.39i)T + 83iT^{2} \)
89 \( 1 + (-5.04 + 8.74i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.97 - 14.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.64328771294492142970958914506, −9.995145803888467894500715963901, −9.261680478409527211747934969599, −8.019551764910083780248990096482, −7.61252197095418955898581467896, −6.44626229786083192378501573529, −5.75380343771273870650251192143, −4.24279426275396430815222195507, −3.36184052535659255360673810054, −1.74188728699983594456029145549, 1.12422471834432666497479063008, 2.29280909382182369500271542166, 3.66120782767711012622546975620, 4.69859789573548443061214599020, 5.72625995032635396700329642665, 7.09618736154145894267196850508, 8.280497992774121482719150416568, 8.747345711644863025515504727541, 9.603360870883463586757391036066, 10.29832551032193664513753853790

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