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Properties

Label 2-24e2-9.4-c1-0-20
Degree $2$
Conductor $576$
Sign $-0.766 + 0.642i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  − 1.73i·3-s + (1.5 − 2.59i)5-s + (−0.5 − 0.866i)7-s − 2.99·9-s + (−1.5 − 2.59i)11-s + (−0.5 + 0.866i)13-s + (−4.5 − 2.59i)15-s + 6·17-s − 4·19-s + (−1.49 + 0.866i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + 5.19i·27-s + (1.5 + 2.59i)29-s + (2.5 − 4.33i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.670 − 1.16i)5-s + (−0.188 − 0.327i)7-s − 0.999·9-s + (−0.452 − 0.783i)11-s + (−0.138 + 0.240i)13-s + (−1.16 − 0.670i)15-s + 1.45·17-s − 0.917·19-s + (−0.327 + 0.188i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + 0.999i·27-s + (0.278 + 0.482i)29-s + (0.449 − 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467137 - 1.28345i\)
\(L(\frac12)\) \(\approx\) \(0.467137 - 1.28345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 1.73iT \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31096860861319511967388779965, −9.410188143575859190711307273622, −8.447248011391483894540889885781, −7.899711178837467910326372702383, −6.69694017851654113679538556225, −5.77046107631856245557002806581, −5.07113302808418267133773026323, −3.46573541860135892652635766635, −1.99699374638027380507059778361, −0.77188997283383854367890158644, 2.38424784963553963887194806924, 3.20140224840601163360589573843, 4.50293706671950974106840890426, 5.60836974882995500954408250760, 6.34856182563885755102335982448, 7.49006187641192219307333481718, 8.566916073724740167392442946972, 9.629299710812925128844189685190, 10.35062920676357758302790618139, 10.54316616454141044819462304066

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