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Properties

Label 2-572-52.47-c1-0-11
Degree $2$
Conductor $572$
Sign $0.668 - 0.743i$
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  + (−1.37 + 0.315i)2-s − 0.826i·3-s + (1.80 − 0.870i)4-s + (1.70 + 1.70i)5-s + (0.260 + 1.13i)6-s + (0.257 + 0.257i)7-s + (−2.20 + 1.76i)8-s + 2.31·9-s + (−2.89 − 1.81i)10-s + (0.707 + 0.707i)11-s + (−0.719 − 1.48i)12-s + (−0.370 + 3.58i)13-s + (−0.436 − 0.273i)14-s + (1.41 − 1.41i)15-s + (2.48 − 3.13i)16-s + 4.38i·17-s + ⋯
L(s)  = 1  + (−0.974 + 0.223i)2-s − 0.477i·3-s + (0.900 − 0.435i)4-s + (0.763 + 0.763i)5-s + (0.106 + 0.465i)6-s + (0.0973 + 0.0973i)7-s + (−0.780 + 0.625i)8-s + 0.772·9-s + (−0.914 − 0.574i)10-s + (0.213 + 0.213i)11-s + (−0.207 − 0.429i)12-s + (−0.102 + 0.994i)13-s + (−0.116 − 0.0731i)14-s + (0.364 − 0.364i)15-s + (0.621 − 0.783i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03260 + 0.460214i\)
\(L(\frac12)\) \(\approx\) \(1.03260 + 0.460214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.315i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.370 - 3.58i)T \)
good3 \( 1 + 0.826iT - 3T^{2} \)
5 \( 1 + (-1.70 - 1.70i)T + 5iT^{2} \)
7 \( 1 + (-0.257 - 0.257i)T + 7iT^{2} \)
17 \( 1 - 4.38iT - 17T^{2} \)
19 \( 1 + (0.588 - 0.588i)T - 19iT^{2} \)
23 \( 1 + 5.90T + 23T^{2} \)
29 \( 1 - 0.751T + 29T^{2} \)
31 \( 1 + (1.32 - 1.32i)T - 31iT^{2} \)
37 \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \)
41 \( 1 + (-6.76 - 6.76i)T + 41iT^{2} \)
43 \( 1 - 2.60T + 43T^{2} \)
47 \( 1 + (-2.12 - 2.12i)T + 47iT^{2} \)
53 \( 1 + 3.28T + 53T^{2} \)
59 \( 1 + (-10.4 - 10.4i)T + 59iT^{2} \)
61 \( 1 - 1.89T + 61T^{2} \)
67 \( 1 + (-0.782 + 0.782i)T - 67iT^{2} \)
71 \( 1 + (2.27 - 2.27i)T - 71iT^{2} \)
73 \( 1 + (4.45 - 4.45i)T - 73iT^{2} \)
79 \( 1 + 7.79iT - 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + (-0.998 + 0.998i)T - 89iT^{2} \)
97 \( 1 + (-0.719 - 0.719i)T + 97iT^{2} \)
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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

−10.52108861802783156159401512679, −10.00109841890980683041932725778, −9.194879879609113376508836176882, −8.129555527565083397578808898229, −7.25140685700054201573323916147, −6.47688822140687766127240528439, −5.90177876436862526737462443960, −4.16345447418036017188303995344, −2.37846561047053682637789512922, −1.59258288139081768335774338736, 0.953955646870829489355951294879, 2.34064950358684205993700396896, 3.79562356854050817014129976214, 5.08187169019641441228146501960, 6.08534920887189511254441916560, 7.28031359485358248813920930673, 8.114046513642554765000248212575, 9.173282367017565949426216823921, 9.651238387561930868719606459140, 10.36373915472217632967400057572

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