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Properties

Label 2-24e2-8.3-c2-0-14
Degree $2$
Conductor $576$
Sign $0.258 + 0.965i$
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  + 3.46i·5-s − 10i·7-s + 13.8·11-s − 6.92i·13-s − 30·17-s − 6.92·19-s + 12i·23-s + 13.0·25-s − 51.9i·29-s − 14i·31-s + 34.6·35-s − 55.4i·37-s + 6·41-s + 62.3·43-s − 84i·47-s + ⋯
L(s)  = 1  + 0.692i·5-s − 1.42i·7-s + 1.25·11-s − 0.532i·13-s − 1.76·17-s − 0.364·19-s + 0.521i·23-s + 0.520·25-s − 1.79i·29-s − 0.451i·31-s + 0.989·35-s − 1.49i·37-s + 0.146·41-s + 1.45·43-s − 1.78i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.541720946\)
\(L(\frac12)\) \(\approx\) \(1.541720946\)
\(L(2)\) 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 \( 1 - 3.46iT - 25T^{2} \)
7 \( 1 + 10iT - 49T^{2} \)
11 \( 1 - 13.8T + 121T^{2} \)
13 \( 1 + 6.92iT - 169T^{2} \)
17 \( 1 + 30T + 289T^{2} \)
19 \( 1 + 6.92T + 361T^{2} \)
23 \( 1 - 12iT - 529T^{2} \)
29 \( 1 + 51.9iT - 841T^{2} \)
31 \( 1 + 14iT - 961T^{2} \)
37 \( 1 + 55.4iT - 1.36e3T^{2} \)
41 \( 1 - 6T + 1.68e3T^{2} \)
43 \( 1 - 62.3T + 1.84e3T^{2} \)
47 \( 1 + 84iT - 2.20e3T^{2} \)
53 \( 1 + 17.3iT - 2.80e3T^{2} \)
59 \( 1 - 62.3T + 3.48e3T^{2} \)
61 \( 1 - 96.9iT - 3.72e3T^{2} \)
67 \( 1 - 48.4T + 4.48e3T^{2} \)
71 \( 1 + 60iT - 5.04e3T^{2} \)
73 \( 1 + 86T + 5.32e3T^{2} \)
79 \( 1 + 38iT - 6.24e3T^{2} \)
83 \( 1 + 13.8T + 6.88e3T^{2} \)
89 \( 1 + 78T + 7.92e3T^{2} \)
97 \( 1 - 62T + 9.40e3T^{2} \)
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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

−10.49028524096563118541734104410, −9.552019412062116224604496767825, −8.606816162038943483748287018081, −7.38893709189746923848017209327, −6.86905918201796305504872498717, −5.96676204210569846680043446906, −4.31880862200366076178991506554, −3.81276735275334532379363049809, −2.27809206313197820495624340122, −0.61283480987577486908157251457, 1.47183051499872596619843961828, 2.69489393388727772336867656866, 4.24792991917049444724925928247, 5.03844751664607348139678014881, 6.24655490795604291462452845787, 6.86218345965042272201075070055, 8.532101527460893961976831888999, 8.859263750765588976718936343055, 9.453411902735921273701493323607, 10.90224244428363022207449041198

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