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Properties

Label 2-24e2-144.85-c1-0-19
Degree $2$
Conductor $576$
Sign $0.975 + 0.218i$
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  + (1.71 + 0.215i)3-s + (2.73 − 0.733i)5-s + (1.14 − 0.660i)7-s + (2.90 + 0.740i)9-s + (0.343 − 1.28i)11-s + (−0.902 − 3.36i)13-s + (4.86 − 0.670i)15-s − 7.60·17-s + (−4.32 + 4.32i)19-s + (2.11 − 0.889i)21-s + (3.46 + 1.99i)23-s + (2.62 − 1.51i)25-s + (4.83 + 1.89i)27-s + (−3.54 − 0.950i)29-s + (−0.569 + 0.985i)31-s + ⋯
L(s)  = 1  + (0.992 + 0.124i)3-s + (1.22 − 0.328i)5-s + (0.432 − 0.249i)7-s + (0.969 + 0.246i)9-s + (0.103 − 0.387i)11-s + (−0.250 − 0.933i)13-s + (1.25 − 0.173i)15-s − 1.84·17-s + (−0.991 + 0.991i)19-s + (0.460 − 0.194i)21-s + (0.721 + 0.416i)23-s + (0.524 − 0.303i)25-s + (0.930 + 0.365i)27-s + (−0.658 − 0.176i)29-s + (−0.102 + 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.975 + 0.218i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44755 - 0.270534i\)
\(L(\frac12)\) \(\approx\) \(2.44755 - 0.270534i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.215i)T \)
good5 \( 1 + (-2.73 + 0.733i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.14 + 0.660i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.343 + 1.28i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.902 + 3.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + (4.32 - 4.32i)T - 19iT^{2} \)
23 \( 1 + (-3.46 - 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 + 0.950i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.569 - 0.985i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.26 - 2.26i)T + 37iT^{2} \)
41 \( 1 + (1.42 + 0.821i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.65 - 6.17i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.58 + 7.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.72 - 7.72i)T + 53iT^{2} \)
59 \( 1 + (-4.80 + 1.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.92 - 2.66i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.73 - 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.87iT - 71T^{2} \)
73 \( 1 - 0.577iT - 73T^{2} \)
79 \( 1 + (0.716 + 1.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.30 + 0.885i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 + (0.648 + 1.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44169108772444399441083535144, −9.791260936422752987288886327113, −8.852909431217937005527475490076, −8.325152614804036729480619237276, −7.19318539266759765897503193176, −6.12126676922459416738999642132, −5.05228542579987949030758791607, −3.99109036812948888562322677018, −2.60269750226084353891408874796, −1.61664188265702931155011730714, 2.02517138913624461832043684793, 2.37928812430972349983403064163, 4.10803523768550179053319309176, 5.04992662786421746116629059829, 6.65224801920989799313980115461, 6.85544048818925832017578809531, 8.359260207758998197502618583648, 9.092509503730448530672055445141, 9.594672155153270653066265936619, 10.67363628290709809104143932534

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