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Properties

Label 2-24e2-72.13-c1-0-8
Degree $2$
Conductor $576$
Sign $0.0337 - 0.999i$
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  + (0.492 + 1.66i)3-s + (1.5 + 0.866i)5-s + (2.17 + 3.77i)7-s + (−2.51 + 1.63i)9-s + (5.05 − 2.91i)11-s + (−3.48 − 2.00i)13-s + (−0.698 + 2.91i)15-s + 1.70·17-s − 2i·19-s + (−5.18 + 5.47i)21-s + (−3.32 + 5.75i)23-s + (−1 − 1.73i)25-s + (−3.95 − 3.36i)27-s + (−4.06 + 2.34i)29-s + (2.43 − 4.21i)31-s + ⋯
L(s)  = 1  + (0.284 + 0.958i)3-s + (0.670 + 0.387i)5-s + (0.822 + 1.42i)7-s + (−0.838 + 0.545i)9-s + (1.52 − 0.879i)11-s + (−0.965 − 0.557i)13-s + (−0.180 + 0.753i)15-s + 0.414·17-s − 0.458i·19-s + (−1.13 + 1.19i)21-s + (−0.692 + 1.19i)23-s + (−0.200 − 0.346i)25-s + (−0.761 − 0.648i)27-s + (−0.754 + 0.435i)29-s + (0.436 − 0.756i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.0337 - 0.999i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.0337 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38396 + 1.33801i\)
\(L(\frac12)\) \(\approx\) \(1.38396 + 1.33801i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.492 - 1.66i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.17 - 3.77i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.05 + 2.91i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.48 + 2.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (3.32 - 5.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.06 - 2.34i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.43 + 4.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.75iT - 37T^{2} \)
41 \( 1 + (-0.646 + 1.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.06 + 1.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.30 - 9.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.506iT - 53T^{2} \)
59 \( 1 + (1.62 + 0.936i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.06 + 0.612i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.36 - 0.789i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.88T + 71T^{2} \)
73 \( 1 + 7.96T + 73T^{2} \)
79 \( 1 + (-1.03 - 1.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.33 - 0.771i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.96T + 89T^{2} \)
97 \( 1 + (3.06 + 5.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.00208931712912868560284279798, −9.825964376975353885152888805011, −9.276154980362279907069192122033, −8.583296569541648600676974186425, −7.55472685285991339089947081109, −5.86768644890332519784638893069, −5.65276224540113087661092843544, −4.34517995733635921354838177339, −3.08948982871837064003996642520, −2.05013313140130541316989733299, 1.21727292637397835746882709753, 2.03434019370722524547757883256, 3.89170360908891357238989622156, 4.78780725560285322653650429624, 6.19037708054231776138123898208, 7.05140325638113994424582278708, 7.63836893053714851223414517361, 8.708172732001045748854623877095, 9.622543086946925380586743683789, 10.36672826639963027405873556354

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