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Properties

Label 2-24e2-24.5-c6-0-29
Degree $2$
Conductor $576$
Sign $0.985 + 0.169i$
Analytic cond. $132.511$
Root an. cond. $11.5113$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.3·5-s + 532.·7-s − 1.84e3·11-s + 1.00e3i·13-s − 3.71e3i·17-s + 7.86e3i·19-s − 1.95e4i·23-s − 1.54e4·25-s + 4.25e4·29-s + 3.91e4·31-s + 7.65e3·35-s + 6.17e4i·37-s + 1.33e4i·41-s − 5.79e4i·43-s − 5.19e4i·47-s + ⋯
L(s)  = 1  + 0.115·5-s + 1.55·7-s − 1.38·11-s + 0.457i·13-s − 0.756i·17-s + 1.14i·19-s − 1.60i·23-s − 0.986·25-s + 1.74·29-s + 1.31·31-s + 0.178·35-s + 1.22i·37-s + 0.193i·41-s − 0.728i·43-s − 0.500i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(132.511\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3),\ 0.985 + 0.169i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.571039573\)
\(L(\frac12)\) \(\approx\) \(2.571039573\)
\(L(4)\) 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 \( 1 \)
good5 \( 1 - 14.3T + 1.56e4T^{2} \)
7 \( 1 - 532.T + 1.17e5T^{2} \)
11 \( 1 + 1.84e3T + 1.77e6T^{2} \)
13 \( 1 - 1.00e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.71e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.86e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.95e4iT - 1.48e8T^{2} \)
29 \( 1 - 4.25e4T + 5.94e8T^{2} \)
31 \( 1 - 3.91e4T + 8.87e8T^{2} \)
37 \( 1 - 6.17e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.33e4iT - 4.75e9T^{2} \)
43 \( 1 + 5.79e4iT - 6.32e9T^{2} \)
47 \( 1 + 5.19e4iT - 1.07e10T^{2} \)
53 \( 1 - 5.29e4T + 2.21e10T^{2} \)
59 \( 1 - 9.58e4T + 4.21e10T^{2} \)
61 \( 1 + 2.48e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.70e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.77e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.81e5T + 1.51e11T^{2} \)
79 \( 1 + 3.58e5T + 2.43e11T^{2} \)
83 \( 1 + 8.58e4T + 3.26e11T^{2} \)
89 \( 1 + 1.09e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.01e6T + 8.32e11T^{2} \)
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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

−10.07558371939667195667305816181, −8.393414218011273881227615729656, −8.279579227156614433398527189779, −7.19303316914296342496907889850, −6.02115184880284230200024729667, −4.95434543146362171157721477517, −4.43497170021612197568036282194, −2.79897053231536507578601847793, −1.91073735610194494594373986661, −0.69469977197798954528029091378, 0.78092513759151890551902328105, 1.92451370101673668423246183104, 2.91641829870176463563786082815, 4.39770984005586197847898250831, 5.13865744057006170663653258319, 5.96439803348293371141712394810, 7.42100774268159639546467721513, 7.973918571850907951796737206572, 8.706225177491650604420363721078, 9.950406359629505758465132350273

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