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Properties

Label 2-572-11.9-c1-0-3
Degree $2$
Conductor $572$
Sign $0.828 - 0.560i$
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  + (−0.0637 + 0.0462i)3-s + (−0.454 − 1.39i)5-s + (1.50 + 1.09i)7-s + (−0.925 + 2.84i)9-s + (1.73 + 2.82i)11-s + (−0.309 + 0.951i)13-s + (0.0936 + 0.0680i)15-s + (0.0702 + 0.216i)17-s + (1.31 − 0.954i)19-s − 0.146·21-s + 3.95·23-s + (2.29 − 1.66i)25-s + (−0.145 − 0.448i)27-s + (6.74 + 4.90i)29-s + (1.39 − 4.28i)31-s + ⋯
L(s)  = 1  + (−0.0367 + 0.0267i)3-s + (−0.203 − 0.625i)5-s + (0.568 + 0.413i)7-s + (−0.308 + 0.949i)9-s + (0.521 + 0.853i)11-s + (−0.0857 + 0.263i)13-s + (0.0241 + 0.0175i)15-s + (0.0170 + 0.0524i)17-s + (0.301 − 0.218i)19-s − 0.0319·21-s + 0.824·23-s + (0.458 − 0.333i)25-s + (−0.0280 − 0.0863i)27-s + (1.25 + 0.910i)29-s + (0.250 − 0.770i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.828 - 0.560i$
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: \((2,\ 572,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40690 + 0.431117i\)
\(L(\frac12)\) \(\approx\) \(1.40690 + 0.431117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2.82i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.0637 - 0.0462i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.454 + 1.39i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.50 - 1.09i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-0.0702 - 0.216i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.31 + 0.954i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.95T + 23T^{2} \)
29 \( 1 + (-6.74 - 4.90i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.39 + 4.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.67 - 4.12i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.20 - 3.05i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.830T + 43T^{2} \)
47 \( 1 + (4.13 - 3.00i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.48 - 13.8i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.35 + 2.43i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.318 + 0.981i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.898T + 67T^{2} \)
71 \( 1 + (1.82 + 5.61i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.00 - 0.730i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.58 + 4.86i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.53 + 7.79i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.23T + 89T^{2} \)
97 \( 1 + (-3.31 + 10.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.92080169721435885478911839640, −9.899515661380185031713259162419, −8.947529542187817390991755021381, −8.253363678807685526657148194216, −7.36572099648424298930666986716, −6.24726034655768307886805647446, −4.85732496526198190290548289969, −4.65726471149985384587667864628, −2.83681563113605679037897615329, −1.49130935977652435055494930065, 0.982323609671655549033333938098, 2.93136006077052083670909991850, 3.77937687711105383454634370641, 5.07144711603583388079750382894, 6.26358822229936266456002170033, 6.94083442526597289990558429126, 8.030797740772605751108446093032, 8.832518014326871799632822272683, 9.821230598287931350141231366317, 10.80887806241213985779082275323

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