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Properties

Label 2-24e2-9.7-c1-0-17
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  + 1.73·3-s + (−0.5 − 0.866i)5-s + (0.866 − 1.5i)7-s + 2.99·9-s + (0.866 − 1.5i)11-s + (−1.5 − 2.59i)13-s + (−0.866 − 1.49i)15-s + 4·17-s − 6.92·19-s + (1.49 − 2.59i)21-s + (4.33 + 7.5i)23-s + (2 − 3.46i)25-s + 5.19·27-s + (0.5 − 0.866i)29-s + (−2.59 − 4.5i)31-s + ⋯
L(s)  = 1  + 1.00·3-s + (−0.223 − 0.387i)5-s + (0.327 − 0.566i)7-s + 0.999·9-s + (0.261 − 0.452i)11-s + (−0.416 − 0.720i)13-s + (−0.223 − 0.387i)15-s + 0.970·17-s − 1.58·19-s + (0.327 − 0.566i)21-s + (0.902 + 1.56i)23-s + (0.400 − 0.692i)25-s + 1.00·27-s + (0.0928 − 0.160i)29-s + (−0.466 − 0.808i)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: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93099 - 0.702823i\)
\(L(\frac12)\) \(\approx\) \(1.93099 - 0.702823i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.866 + 1.5i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (-4.33 - 7.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.59 + 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.33 - 7.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 + 7.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-2.59 + 4.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.33 - 7.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.50153886123028986620237201452, −9.670137959497789114141263133758, −8.791993175268844883188276758470, −7.939256226723225574227808044498, −7.42316762803947496260423256939, −6.10719887708062703033876344820, −4.77990081670115806645181361534, −3.87577459947347825383008362617, −2.78894453054846726077968288434, −1.21157499120624372355627940872, 1.84892681640977645116623063046, 2.88473684136154877022035332505, 4.08702260743583094319812045499, 5.04661427229553903000466013317, 6.62946937773544826787486948843, 7.20537508240674074125329787778, 8.426577884207728553476667439899, 8.831749785372098941619287289100, 9.910268533903779458466732579461, 10.65405639227431787989543446022

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