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Properties

Label 2-24e2-16.3-c2-0-5
Degree $2$
Conductor $576$
Sign $0.718 - 0.695i$
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  + (0.0586 − 0.0586i)5-s − 4.61·7-s + (−5.36 − 5.36i)11-s + (11.0 + 11.0i)13-s + 12.8·17-s + (−2.63 + 2.63i)19-s + 16.3·23-s + 24.9i·25-s + (26.0 + 26.0i)29-s + 20.2i·31-s + (−0.270 + 0.270i)35-s + (41.2 − 41.2i)37-s + 3.29i·41-s + (0.786 + 0.786i)43-s + 79.7i·47-s + ⋯
L(s)  = 1  + (0.0117 − 0.0117i)5-s − 0.659·7-s + (−0.487 − 0.487i)11-s + (0.850 + 0.850i)13-s + 0.757·17-s + (−0.138 + 0.138i)19-s + 0.712·23-s + 0.999i·25-s + (0.898 + 0.898i)29-s + 0.652i·31-s + (−0.00773 + 0.00773i)35-s + (1.11 − 1.11i)37-s + 0.0804i·41-s + (0.0183 + 0.0183i)43-s + 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.571833030\)
\(L(\frac12)\) \(\approx\) \(1.571833030\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-0.0586 + 0.0586i)T - 25iT^{2} \)
7 \( 1 + 4.61T + 49T^{2} \)
11 \( 1 + (5.36 + 5.36i)T + 121iT^{2} \)
13 \( 1 + (-11.0 - 11.0i)T + 169iT^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 + (2.63 - 2.63i)T - 361iT^{2} \)
23 \( 1 - 16.3T + 529T^{2} \)
29 \( 1 + (-26.0 - 26.0i)T + 841iT^{2} \)
31 \( 1 - 20.2iT - 961T^{2} \)
37 \( 1 + (-41.2 + 41.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (-0.786 - 0.786i)T + 1.84e3iT^{2} \)
47 \( 1 - 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (1.06 - 1.06i)T - 2.80e3iT^{2} \)
59 \( 1 + (-32.5 - 32.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (-15.2 - 15.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (-60.0 + 60.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 56.3T + 5.04e3T^{2} \)
73 \( 1 - 9.70iT - 5.32e3T^{2} \)
79 \( 1 - 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (-26.7 + 26.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−10.75172189160716524366357308620, −9.651039942024546932686819543302, −8.961493550392697247960745024225, −8.011392121942940518530534329781, −6.95109576550565625512823268809, −6.11270980693072335927940788742, −5.12695958270918192875444093631, −3.80911123357718729674821396577, −2.86938672587295079393994961783, −1.17641299164222942036194619171, 0.71320996049476817293713459482, 2.52556799116234180940655182027, 3.56745857314693903701219736404, 4.81511620378430096949516914513, 5.90206848015050875026879830147, 6.70580608865927781607147740271, 7.86605905607717198045519794042, 8.519551311215669285749383682692, 9.810000009793122757348061403063, 10.19240467100185427742522058568

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