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Properties

Label 2-572-572.51-c1-0-69
Degree $2$
Conductor $572$
Sign $-0.811 + 0.584i$
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  + (0.831 − 1.14i)2-s + (−0.618 − 1.90i)4-s + (1.76 − 0.572i)7-s + (−2.68 − 0.874i)8-s + (−2.42 − 1.76i)9-s + (−2.80 − 1.76i)11-s + (2.11 − 2.91i)13-s + (0.809 − 2.49i)14-s + (−3.23 + 2.35i)16-s + (2.04 + 2.81i)17-s + (−4.03 + 1.31i)18-s + (−2.53 − 0.823i)19-s + (−4.35 + 1.73i)22-s + (1.54 − 4.75i)25-s + (−1.57 − 4.84i)26-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.666 − 0.216i)7-s + (−0.951 − 0.309i)8-s + (−0.809 − 0.587i)9-s + (−0.845 − 0.533i)11-s + (0.587 − 0.809i)13-s + (0.216 − 0.666i)14-s + (−0.809 + 0.587i)16-s + (0.496 + 0.683i)17-s + (−0.951 + 0.309i)18-s + (−0.581 − 0.188i)19-s + (−0.928 + 0.370i)22-s + (0.309 − 0.951i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504877 - 1.56414i\)
\(L(\frac12)\) \(\approx\) \(0.504877 - 1.56414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 1.14i)T \)
11 \( 1 + (2.80 + 1.76i)T \)
13 \( 1 + (-2.11 + 2.91i)T \)
good3 \( 1 + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.76 + 0.572i)T + (5.66 - 4.11i)T^{2} \)
17 \( 1 + (-2.04 - 2.81i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.53 + 0.823i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-9.35 + 3.03i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.34 + 4.60i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (3.58 - 11.0i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.5 - 8.42i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.87 + 5.77i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.734 - 1.01i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + (-13.3 + 9.71i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.3 - 14.2i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 - 92.2i)T^{2} \)
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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

−10.79411792845490014375525651000, −9.766783478234710237629456911509, −8.567608162806476658917531048286, −8.032805225307142724331278952986, −6.31058467137500785404282080392, −5.69375096202674348103380344307, −4.62296015812725819274593729727, −3.48331702539088157032940209753, −2.49694062169786424917855063749, −0.77774249706653970067887936161, 2.25504461410124498494951260760, 3.53444721938072216597236980374, 4.98168108155821952567925851467, 5.27183242395164802349513111305, 6.58774126751896069103392106473, 7.44613003185059445058545253729, 8.392457501726313833690698649530, 8.880685869997211449375504784460, 10.27574204166479042711486813045, 11.29889058785972133254188403799

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