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Properties

Label 2-572-13.4-c1-0-11
Degree $2$
Conductor $572$
Sign $-0.467 + 0.883i$
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  + (1.46 − 2.52i)3-s − 2.68i·5-s + (2.06 − 1.18i)7-s + (−2.76 − 4.79i)9-s + (0.866 + 0.5i)11-s + (1.87 + 3.07i)13-s + (−6.78 − 3.91i)15-s + (1.66 + 2.88i)17-s + (−2.87 + 1.66i)19-s − 6.95i·21-s + (−1.45 + 2.51i)23-s − 2.18·25-s − 7.40·27-s + (0.873 − 1.51i)29-s + 9.63i·31-s + ⋯
L(s)  = 1  + (0.843 − 1.46i)3-s − 1.19i·5-s + (0.778 − 0.449i)7-s + (−0.922 − 1.59i)9-s + (0.261 + 0.150i)11-s + (0.520 + 0.853i)13-s + (−1.75 − 1.01i)15-s + (0.404 + 0.700i)17-s + (−0.660 + 0.381i)19-s − 1.51i·21-s + (−0.302 + 0.524i)23-s − 0.437·25-s − 1.42·27-s + (0.162 − 0.281i)29-s + 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.467 + 0.883i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.467 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06539 - 1.76870i\)
\(L(\frac12)\) \(\approx\) \(1.06539 - 1.76870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-1.87 - 3.07i)T \)
good3 \( 1 + (-1.46 + 2.52i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.68iT - 5T^{2} \)
7 \( 1 + (-2.06 + 1.18i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (-1.66 - 2.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.87 - 1.66i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.45 - 2.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.873 + 1.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.63iT - 31T^{2} \)
37 \( 1 + (2.49 + 1.44i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.25 - 1.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.85 + 3.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.65iT - 47T^{2} \)
53 \( 1 + 7.18T + 53T^{2} \)
59 \( 1 + (4.96 - 2.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.76 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.52 + 4.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.6 + 6.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.84iT - 73T^{2} \)
79 \( 1 - 2.33T + 79T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + (3.99 + 2.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.72 - 5.03i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.45050404111335372238425146747, −9.075247120076761584628944878091, −8.565946127501562010750480929747, −7.924356140426847892535658881938, −7.03256561298396321503181504272, −6.06745198257250828700434806675, −4.74303846081125436110510204571, −3.63048825647111581342175174150, −1.84998793460344392344034990405, −1.26495634459292788956458468813, 2.45862301327436347662786828132, 3.23581767712730985441814413504, 4.26338023619346198161369116291, 5.28490195035713961841045025555, 6.42053586876503819655741773065, 7.80180097861617597676090802982, 8.419318591454129028451600218538, 9.407792630473062640617544177611, 10.10262167962559426060570921231, 11.02835753910220717192814971882

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