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Properties

Label 2-24e2-9.5-c2-0-24
Degree $2$
Conductor $576$
Sign $0.586 + 0.809i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.106 − 2.99i)3-s + (1.83 + 1.06i)5-s + (3.42 + 5.93i)7-s + (−8.97 − 0.640i)9-s + (−4.54 + 2.62i)11-s + (10.0 − 17.3i)13-s + (3.38 − 5.40i)15-s + 0.0666i·17-s + 27.2·19-s + (18.1 − 9.63i)21-s + (18.5 + 10.7i)23-s + (−10.2 − 17.7i)25-s + (−2.87 + 26.8i)27-s + (37.2 − 21.5i)29-s + (−8.06 + 13.9i)31-s + ⋯
L(s)  = 1  + (0.0356 − 0.999i)3-s + (0.367 + 0.212i)5-s + (0.489 + 0.847i)7-s + (−0.997 − 0.0711i)9-s + (−0.413 + 0.238i)11-s + (0.770 − 1.33i)13-s + (0.225 − 0.360i)15-s + 0.00392i·17-s + 1.43·19-s + (0.864 − 0.458i)21-s + (0.807 + 0.466i)23-s + (−0.409 − 0.709i)25-s + (−0.106 + 0.994i)27-s + (1.28 − 0.742i)29-s + (−0.260 + 0.450i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.586 + 0.809i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.586 + 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.043577010\)
\(L(\frac12)\) \(\approx\) \(2.043577010\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.106 + 2.99i)T \)
good5 \( 1 + (-1.83 - 1.06i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.42 - 5.93i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.54 - 2.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.0 + 17.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 0.0666iT - 289T^{2} \)
19 \( 1 - 27.2T + 361T^{2} \)
23 \( 1 + (-18.5 - 10.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-37.2 + 21.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (8.06 - 13.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 22.3T + 1.36e3T^{2} \)
41 \( 1 + (2.56 + 1.48i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (14.6 + 25.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-38.3 + 22.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 12.8iT - 2.80e3T^{2} \)
59 \( 1 + (75.0 + 43.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.45 + 11.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-61.4 + 106. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 - 90.7T + 5.32e3T^{2} \)
79 \( 1 + (-24.2 - 42.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (101. - 58.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 116. iT - 7.92e3T^{2} \)
97 \( 1 + (23.2 + 40.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.47628395090163586435058987425, −9.427313218800120014500764125148, −8.345442671255185502195502939687, −7.87181188631952727650196371136, −6.77074643681996985673715684715, −5.74400509742010354067270575030, −5.19800916084386062084031510884, −3.22919756395402004104486871318, −2.32075456872554048094271353916, −0.952488186198548823411073231514, 1.22723324482679191610823912440, 2.98140840326563792902481448458, 4.12596176640771670478676034871, 4.90696895364564271116015387307, 5.89719798850846193679312256636, 7.08232359798382649269554073847, 8.152691011916206412423431546776, 9.105596342749499816262993467844, 9.685750188458949670580929033949, 10.77863096220156390089044629289

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