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Properties

Label 2-574-287.100-c1-0-10
Degree $2$
Conductor $574$
Sign $0.779 - 0.626i$
Analytic cond. $4.58341$
Root an. cond. $2.14089$
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  + (0.978 + 0.207i)2-s + (−0.523 − 0.906i)3-s + (0.913 + 0.406i)4-s + (0.0626 + 0.596i)5-s + (−0.323 − 0.995i)6-s + (0.226 + 2.63i)7-s + (0.809 + 0.587i)8-s + (0.951 − 1.64i)9-s + (−0.0626 + 0.596i)10-s + (−0.605 + 5.76i)11-s + (−0.109 − 1.04i)12-s + (0.807 + 2.48i)13-s + (−0.326 + 2.62i)14-s + (0.507 − 0.368i)15-s + (0.669 + 0.743i)16-s + (0.442 − 4.20i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.147i)2-s + (−0.302 − 0.523i)3-s + (0.456 + 0.203i)4-s + (0.0280 + 0.266i)5-s + (−0.132 − 0.406i)6-s + (0.0857 + 0.996i)7-s + (0.286 + 0.207i)8-s + (0.317 − 0.549i)9-s + (−0.0198 + 0.188i)10-s + (−0.182 + 1.73i)11-s + (−0.0315 − 0.300i)12-s + (0.223 + 0.689i)13-s + (−0.0871 + 0.701i)14-s + (0.131 − 0.0952i)15-s + (0.167 + 0.185i)16-s + (0.107 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(4.58341\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{574} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ 0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95156 + 0.686627i\)
\(L(\frac12)\) \(\approx\) \(1.95156 + 0.686627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (-0.226 - 2.63i)T \)
41 \( 1 + (-0.732 - 6.36i)T \)
good3 \( 1 + (0.523 + 0.906i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.0626 - 0.596i)T + (-4.89 + 1.03i)T^{2} \)
11 \( 1 + (0.605 - 5.76i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (-0.807 - 2.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.442 + 4.20i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-0.741 - 0.823i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (-4.29 - 0.912i)T + (21.0 + 9.35i)T^{2} \)
29 \( 1 + (1.04 - 0.762i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.700 + 6.66i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (0.0806 + 0.767i)T + (-36.1 + 7.69i)T^{2} \)
43 \( 1 + (-1.32 - 4.08i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.584 - 0.124i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (11.2 + 4.99i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-5.17 + 5.75i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (4.98 + 5.53i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-13.7 - 6.14i)T + (44.8 + 49.7i)T^{2} \)
71 \( 1 + (2.08 + 1.51i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.43 + 9.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.75 - 3.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.17T + 83T^{2} \)
89 \( 1 + (3.50 + 3.89i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (-10.5 + 7.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.27730019059253219754290607456, −9.787882341250934252996357337051, −9.262898896376601974360154410356, −7.83260437193579790184679874955, −6.97434412426061802046022432537, −6.40956414268392259969239115924, −5.23719079289559722715285575296, −4.40978666655860683882667691394, −2.91630865862976814406945470457, −1.77071566738384651991640973070, 1.10718885820409111901978566339, 3.09401970704944524517424980904, 3.97674852640224245321338364501, 5.03547227052399091701348065438, 5.74542117169014912314386190706, 6.88989602219044767667593887866, 7.961391243794797989405245275957, 8.829233657017965962853202977186, 10.22356617339315549980935672931, 10.83077491445149672917278185456

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