Properties

Label 2-570-95.8-c1-0-3
Degree $2$
Conductor $570$
Sign $0.459 - 0.888i$
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  + (0.965 − 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (−1.40 + 1.74i)5-s + (−0.499 − 0.866i)6-s + (−3.60 + 3.60i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.901 + 2.04i)10-s + 2.76·11-s + (−0.707 − 0.707i)12-s + (5.13 + 1.37i)13-s + (−2.54 + 4.41i)14-s + (2.04 + 0.901i)15-s + (0.500 − 0.866i)16-s + (1.05 + 3.93i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.626 + 0.779i)5-s + (−0.204 − 0.353i)6-s + (−1.36 + 1.36i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.285 + 0.647i)10-s + 0.832·11-s + (−0.204 − 0.204i)12-s + (1.42 + 0.381i)13-s + (−0.681 + 1.18i)14-s + (0.528 + 0.232i)15-s + (0.125 − 0.216i)16-s + (0.255 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.459 - 0.888i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27710 + 0.776855i\)
\(L(\frac12)\) \(\approx\) \(1.27710 + 0.776855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (1.40 - 1.74i)T \)
19 \( 1 + (4.32 - 0.570i)T \)
good7 \( 1 + (3.60 - 3.60i)T - 7iT^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
13 \( 1 + (-5.13 - 1.37i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.05 - 3.93i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.73 - 6.48i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.54 - 2.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.810iT - 31T^{2} \)
37 \( 1 + (-2.19 - 2.19i)T + 37iT^{2} \)
41 \( 1 + (9.40 + 5.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.1 + 2.71i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (9.46 + 2.53i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.41 - 1.18i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.60 - 4.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.92 + 5.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.536 - 2.00i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.86 - 2.80i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.5 + 3.08i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.78 - 6.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.75 + 3.75i)T + 83iT^{2} \)
89 \( 1 + (-2.32 - 4.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.4 + 4.15i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.17366397918205785695406175632, −10.22116507434788861357149767397, −9.060769905073124954111667163037, −8.263191079607143729276458951602, −6.84211511413851844273453388076, −6.32388495325434546368389840602, −5.71922897828493007979225950062, −3.85744194965674726795300532393, −3.27541755463821161104988274159, −1.91756619088636271092965945341, 0.70754535084865349154057806823, 3.25466179280875659464298337911, 4.00104069830874448434897801976, 4.62093348681225408672526540013, 6.13409797859837206408368362924, 6.66988695081840149402878860365, 7.86677214436103989134355065402, 8.841979989856730120318542531650, 9.786922961730463261406464843603, 10.69155565523090483042782994129

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