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Properties

Label 2-572-11.5-c1-0-4
Degree $2$
Conductor $572$
Sign $-0.295 - 0.955i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (2.49 + 1.81i)3-s + (−1.12 + 3.45i)5-s + (1.36 − 0.989i)7-s + (2.01 + 6.18i)9-s + (−3.12 + 1.12i)11-s + (−0.309 − 0.951i)13-s + (−9.07 + 6.59i)15-s + (1.71 − 5.26i)17-s + (−3.91 − 2.84i)19-s + 5.18·21-s + 7.77·23-s + (−6.65 − 4.83i)25-s + (−3.34 + 10.2i)27-s + (2.51 − 1.82i)29-s + (0.814 + 2.50i)31-s + ⋯
L(s)  = 1  + (1.44 + 1.04i)3-s + (−0.502 + 1.54i)5-s + (0.514 − 0.373i)7-s + (0.670 + 2.06i)9-s + (−0.940 + 0.338i)11-s + (−0.0857 − 0.263i)13-s + (−2.34 + 1.70i)15-s + (0.415 − 1.27i)17-s + (−0.897 − 0.651i)19-s + 1.13·21-s + 1.62·23-s + (−1.33 − 0.966i)25-s + (−0.643 + 1.98i)27-s + (0.467 − 0.339i)29-s + (0.146 + 0.450i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.295 - 0.955i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27088 + 1.72248i\)
\(L(\frac12)\) \(\approx\) \(1.27088 + 1.72248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.12 - 1.12i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-2.49 - 1.81i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.12 - 3.45i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.36 + 0.989i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-1.71 + 5.26i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.91 + 2.84i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.77T + 23T^{2} \)
29 \( 1 + (-2.51 + 1.82i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.814 - 2.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.27 + 1.65i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.59 - 5.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.33T + 43T^{2} \)
47 \( 1 + (-0.386 - 0.280i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.90 - 5.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.62 + 6.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.18 + 6.72i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.78T + 67T^{2} \)
71 \( 1 + (0.714 - 2.19i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.82 - 7.13i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.715 - 2.20i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.94 - 9.06i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.83T + 89T^{2} \)
97 \( 1 + (2.79 + 8.60i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.80591353704714956760234337365, −10.15987855261664713066619314275, −9.353878411685269713811634957081, −8.264582046889151870433507693226, −7.57618630359846470919535737491, −6.90464644178300630822927102902, −5.04365169541377821403190313782, −4.21512235801146988018313567525, −2.92755492793525871195791460823, −2.68742634945456402790861152793, 1.13839050367406520409885722789, 2.23216109979426273923602551901, 3.58820502669976750460565533151, 4.71814093714193627988934278214, 5.89740536875924088957867975627, 7.28922668648066058804062587181, 8.114841295913516260375503921261, 8.545254814250099099198966100148, 9.017658981177349706914101182738, 10.33377037614021812057171448131

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