Properties

Label 2-24e2-36.31-c2-0-17
Degree $2$
Conductor $576$
Sign $-0.173 - 0.984i$
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  + (1.5 + 2.59i)3-s + (2 − 3.46i)5-s + (−3 + 1.73i)7-s + (−4.5 + 7.79i)9-s + (10.5 − 6.06i)11-s + (−11 + 19.0i)13-s + 12·15-s − 11·17-s + 15.5i·19-s + (−9 − 5.19i)21-s + (21 + 12.1i)23-s + (4.50 + 7.79i)25-s − 27·27-s + (17 + 29.4i)29-s + (6 + 3.46i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.400 − 0.692i)5-s + (−0.428 + 0.247i)7-s + (−0.5 + 0.866i)9-s + (0.954 − 0.551i)11-s + (−0.846 + 1.46i)13-s + 0.800·15-s − 0.647·17-s + 0.820i·19-s + (−0.428 − 0.247i)21-s + (0.913 + 0.527i)23-s + (0.180 + 0.311i)25-s − 27-s + (0.586 + 1.01i)29-s + (0.193 + 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.898957742\)
\(L(\frac12)\) \(\approx\) \(1.898957742\)
\(L(2)\) 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 \( 1 + (-1.5 - 2.59i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.5 + 6.06i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (11 - 19.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 11T + 289T^{2} \)
19 \( 1 - 15.5iT - 361T^{2} \)
23 \( 1 + (-21 - 12.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-17 - 29.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-6 - 3.46i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 16T + 1.36e3T^{2} \)
41 \( 1 + (6.5 - 11.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-43.5 + 25.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-3 + 1.73i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 52T + 2.80e3T^{2} \)
59 \( 1 + (-46.5 - 26.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8 + 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (100.5 + 58.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (-24 + 13.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (30 - 17.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 - 37.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.67131042467780876694849357792, −9.434593469402693689697500090621, −9.271582220201547746820144128472, −8.522679816181749486087993032056, −7.16882119659955852753197577307, −6.11343309719005821825759061812, −4.99972353484572109699649994401, −4.19897685884743815162777634502, −3.04533994279578507370675023386, −1.64109426373387001252958732070, 0.68209706038101936001976696954, 2.37609392985968649441214254318, 3.07386324797668958648397166876, 4.54627221717983970923181519309, 6.03676130456327532352822846911, 6.77831933661985435891676493317, 7.39386228147823790832508474072, 8.459043786487166348478562994941, 9.432382630682692018336566386525, 10.15206846548563057043079508699

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