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Properties

Label 2-572-11.9-c1-0-0
Degree $2$
Conductor $572$
Sign $-0.814 - 0.579i$
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  + (−2.38 + 1.73i)3-s + (−1.16 − 3.59i)5-s + (2.62 + 1.90i)7-s + (1.75 − 5.41i)9-s + (−2.04 + 2.61i)11-s + (0.309 − 0.951i)13-s + (9.01 + 6.55i)15-s + (1.62 + 5.00i)17-s + (−0.605 + 0.439i)19-s − 9.58·21-s − 5.29·23-s + (−7.51 + 5.46i)25-s + (2.45 + 7.55i)27-s + (−6.54 − 4.75i)29-s + (−1.77 + 5.47i)31-s + ⋯
L(s)  = 1  + (−1.37 + 1.00i)3-s + (−0.522 − 1.60i)5-s + (0.993 + 0.721i)7-s + (0.586 − 1.80i)9-s + (−0.616 + 0.787i)11-s + (0.0857 − 0.263i)13-s + (2.32 + 1.69i)15-s + (0.394 + 1.21i)17-s + (−0.138 + 0.100i)19-s − 2.09·21-s − 1.10·23-s + (−1.50 + 1.09i)25-s + (0.472 + 1.45i)27-s + (−1.21 − 0.882i)29-s + (−0.319 + 0.983i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.814 - 0.579i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.814 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124323 + 0.389235i\)
\(L(\frac12)\) \(\approx\) \(0.124323 + 0.389235i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 2.61i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (2.38 - 1.73i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.16 + 3.59i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.62 - 1.90i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-1.62 - 5.00i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.605 - 0.439i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + (6.54 + 4.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.77 - 5.47i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.20 - 4.50i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.15 - 1.56i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.50T + 43T^{2} \)
47 \( 1 + (5.49 - 3.99i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.62 - 11.1i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.191 + 0.139i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.82 - 5.60i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 + (-1.49 - 4.61i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.28 - 6.74i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.441 + 1.35i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.95 + 9.10i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 7.55T + 89T^{2} \)
97 \( 1 + (0.828 - 2.54i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.21247059424307056299028352140, −10.24341904197153840644950808510, −9.496870328778622266558395057122, −8.421168346913665928417316642522, −7.83419933580760567299350947499, −6.02683242955972658054255760019, −5.35326278917765246994395750595, −4.69957140864306025288560450179, −4.01780325092059012726003877113, −1.55104856487725656861402360182, 0.28118741884041545759757373157, 2.04660887441372865000970872292, 3.57626234632707711414576362593, 4.99867573214446855046815494605, 5.96226105587020390755368866258, 6.85750446897683484364859179736, 7.50599233104657223679468500105, 8.020125455803904838866644491717, 9.939769861927037715872858532835, 10.88012737377175746345889343905

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