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Properties

Label 2-24e2-144.11-c1-0-2
Degree $2$
Conductor $576$
Sign $-0.317 - 0.948i$
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  + (−1.10 − 1.33i)3-s + (0.664 + 0.178i)5-s + (−0.645 + 1.11i)7-s + (−0.559 + 2.94i)9-s + (−3.21 + 0.860i)11-s + (−4.74 − 1.27i)13-s + (−0.496 − 1.08i)15-s + 5.58i·17-s + (2.49 + 2.49i)19-s + (2.20 − 0.374i)21-s + (2.36 − 1.36i)23-s + (−3.92 − 2.26i)25-s + (4.55 − 2.50i)27-s + (−2.95 + 0.792i)29-s + (−5.28 + 3.04i)31-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)3-s + (0.297 + 0.0796i)5-s + (−0.244 + 0.422i)7-s + (−0.186 + 0.982i)9-s + (−0.968 + 0.259i)11-s + (−1.31 − 0.352i)13-s + (−0.128 − 0.279i)15-s + 1.35i·17-s + (0.572 + 0.572i)19-s + (0.481 − 0.0816i)21-s + (0.493 − 0.284i)23-s + (−0.784 − 0.452i)25-s + (0.875 − 0.482i)27-s + (−0.549 + 0.147i)29-s + (−0.948 + 0.547i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276740 + 0.384394i\)
\(L(\frac12)\) \(\approx\) \(0.276740 + 0.384394i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 + 1.33i)T \)
good5 \( 1 + (-0.664 - 0.178i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.645 - 1.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.21 - 0.860i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.74 + 1.27i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.58iT - 17T^{2} \)
19 \( 1 + (-2.49 - 2.49i)T + 19iT^{2} \)
23 \( 1 + (-2.36 + 1.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.95 - 0.792i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.28 - 3.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.507 - 0.507i)T + 37iT^{2} \)
41 \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.254 + 0.949i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.13 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.601 - 0.601i)T - 53iT^{2} \)
59 \( 1 + (1.28 - 4.77i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.90 + 10.8i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.0295 + 0.110i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.0447iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + (2.50 + 1.44i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.01 + 3.79i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-4.41 + 7.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92287638513925427001341538765, −10.26082727760225332475866572755, −9.359256878933402586749791536773, −7.977716558738649582213697428268, −7.55030299542607653930684323769, −6.32553429203526095742185160876, −5.62669281604468194922674662001, −4.72309288696301278914133425816, −2.91551123296988078568693419837, −1.80598907994193114919978946659, 0.27188773115805330309428718154, 2.56711870006647489275631791732, 3.83072287225137129097003817188, 5.11257178830235746007435796852, 5.45829996625185494105742210398, 6.93242728716419834367660298742, 7.55277470152658963291393662338, 9.132490397908880945201795359748, 9.626219538994120958831450497095, 10.37990947561736189924297317120

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