Properties

Label 2-572-13.12-c1-0-4
Degree $2$
Conductor $572$
Sign $0.523 - 0.851i$
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  + 1.25·3-s + 2.18i·5-s − 0.251i·7-s − 1.43·9-s + i·11-s + (3.07 + 1.88i)13-s + 2.73i·15-s + 5.77·17-s + 7.21i·19-s − 0.314i·21-s + 1.93·23-s + 0.234·25-s − 5.54·27-s − 5.64·29-s − 3.81i·31-s + ⋯
L(s)  = 1  + 0.722·3-s + 0.976i·5-s − 0.0949i·7-s − 0.478·9-s + 0.301i·11-s + (0.851 + 0.523i)13-s + 0.705i·15-s + 1.40·17-s + 1.65i·19-s − 0.0686i·21-s + 0.402·23-s + 0.0468·25-s − 1.06·27-s − 1.04·29-s − 0.685i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.523 - 0.851i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.523 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56420 + 0.874494i\)
\(L(\frac12)\) \(\approx\) \(1.56420 + 0.874494i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.07 - 1.88i)T \)
good3 \( 1 - 1.25T + 3T^{2} \)
5 \( 1 - 2.18iT - 5T^{2} \)
7 \( 1 + 0.251iT - 7T^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 - 7.21iT - 19T^{2} \)
23 \( 1 - 1.93T + 23T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 + 3.81iT - 31T^{2} \)
37 \( 1 - 6.68iT - 37T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 - 6.50iT - 47T^{2} \)
53 \( 1 - 9.98T + 53T^{2} \)
59 \( 1 + 13.6iT - 59T^{2} \)
61 \( 1 + 9.64T + 61T^{2} \)
67 \( 1 + 11.7iT - 67T^{2} \)
71 \( 1 + 8.73iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 6.39iT - 83T^{2} \)
89 \( 1 + 6.82iT - 89T^{2} \)
97 \( 1 + 10.1iT - 97T^{2} \)
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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

−10.78437634080783801715279964650, −10.01201216671627469904394198156, −9.127818592920123652064706831345, −8.130103810983504434721738194534, −7.47581113819197070910204405734, −6.36774714289204765774356299535, −5.50077292652449555561146087039, −3.79669096866697468628436107133, −3.20172384058522593607306585582, −1.84221215028483669303900888342, 1.03017185286626930310215862552, 2.74679088089360158355478735013, 3.71099589811184224271222357323, 5.09521446856649356401784199415, 5.78691567435165958869941776841, 7.19442194077791475795125475601, 8.212587385719606499652360097989, 8.796193044101980156019184319216, 9.386567541717340679474287184007, 10.59302379091554233794377308268

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