Properties

Label 2-24e2-144.13-c1-0-17
Degree $2$
Conductor $576$
Sign $-0.438 + 0.898i$
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  + (−0.215 + 1.71i)3-s + (−0.733 − 2.73i)5-s + (−1.14 + 0.660i)7-s + (−2.90 − 0.740i)9-s + (−1.28 − 0.343i)11-s + (3.36 − 0.902i)13-s + (4.86 − 0.670i)15-s − 7.60·17-s + (−4.32 − 4.32i)19-s + (−0.889 − 2.11i)21-s + (−3.46 − 1.99i)23-s + (−2.62 + 1.51i)25-s + (1.89 − 4.83i)27-s + (0.950 − 3.54i)29-s + (−0.569 + 0.985i)31-s + ⋯
L(s)  = 1  + (−0.124 + 0.992i)3-s + (−0.328 − 1.22i)5-s + (−0.432 + 0.249i)7-s + (−0.969 − 0.246i)9-s + (−0.387 − 0.103i)11-s + (0.933 − 0.250i)13-s + (1.25 − 0.173i)15-s − 1.84·17-s + (−0.991 − 0.991i)19-s + (−0.194 − 0.460i)21-s + (−0.721 − 0.416i)23-s + (−0.524 + 0.303i)25-s + (0.365 − 0.930i)27-s + (0.176 − 0.658i)29-s + (−0.102 + 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.438 + 0.898i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258626 - 0.413703i\)
\(L(\frac12)\) \(\approx\) \(0.258626 - 0.413703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.215 - 1.71i)T \)
good5 \( 1 + (0.733 + 2.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.14 - 0.660i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.28 + 0.343i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.36 + 0.902i)T + (11.2 - 6.5i)T^{2} \)
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 \( 1 + (-0.950 + 3.54i)T + (-25.1 - 14.5i)T^{2} \)
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 \( 1 + (-6.17 - 1.65i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.58 + 7.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.72 + 7.72i)T - 53iT^{2} \)
59 \( 1 + (1.28 + 4.80i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.66 - 9.92i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (13.9 - 3.73i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-0.885 + 3.30i)T + (-71.8 - 41.5i)T^{2} \)
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

−10.52984245293877809342792000596, −9.382020621192270866281809665425, −8.733545886528681799426938092797, −8.276915653755236727194350350019, −6.59048354757284149309549639275, −5.69691196429754635843613456563, −4.59084036603775651117408295342, −4.07079044186868434082533223847, −2.53815341656034486366070344498, −0.26291228852664394521696331486, 1.94099980255223147432272390673, 3.07816299001060839588132589360, 4.23798212555534039002350735927, 6.01790112117443255736474990602, 6.49791620928128536357880349607, 7.30508528838242957073785173310, 8.166711852357769170491487950254, 9.111038803010170348066095884198, 10.56408370575173722749936133074, 10.89843113626113458967785536659

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