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Properties

Label 2-24e2-36.7-c2-0-8
Degree $2$
Conductor $576$
Sign $-0.0769 - 0.997i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.76 − 1.16i)3-s + (4.03 + 6.98i)5-s + (3.90 + 2.25i)7-s + (6.29 + 6.42i)9-s + (3.25 + 1.88i)11-s + (3.52 + 6.10i)13-s + (−3.03 − 23.9i)15-s + 0.517·17-s − 16.4i·19-s + (−8.17 − 10.7i)21-s + (−27.7 + 15.9i)23-s + (−19.9 + 34.6i)25-s + (−9.94 − 25.1i)27-s + (−9.48 + 16.4i)29-s + (13.1 − 7.58i)31-s + ⋯
L(s)  = 1  + (−0.921 − 0.387i)3-s + (0.806 + 1.39i)5-s + (0.557 + 0.321i)7-s + (0.699 + 0.714i)9-s + (0.296 + 0.171i)11-s + (0.271 + 0.469i)13-s + (−0.202 − 1.59i)15-s + 0.0304·17-s − 0.864i·19-s + (−0.389 − 0.512i)21-s + (−1.20 + 0.695i)23-s + (−0.799 + 1.38i)25-s + (−0.368 − 0.929i)27-s + (−0.327 + 0.566i)29-s + (0.423 − 0.244i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0769 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0769 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.0769 - 0.997i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.0769 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.453672878\)
\(L(\frac12)\) \(\approx\) \(1.453672878\)
\(L(2)\) 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 \( 1 + (2.76 + 1.16i)T \)
good5 \( 1 + (-4.03 - 6.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.90 - 2.25i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.25 - 1.88i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.52 - 6.10i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 0.517T + 289T^{2} \)
19 \( 1 + 16.4iT - 361T^{2} \)
23 \( 1 + (27.7 - 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (9.48 - 16.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-13.1 + 7.58i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 0.592T + 1.36e3T^{2} \)
41 \( 1 + (-12.3 - 21.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-27.8 - 16.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.4 - 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 0.664T + 2.80e3T^{2} \)
59 \( 1 + (30.5 - 17.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.7 - 58.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (74.4 - 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + (126. + 73.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-87.1 - 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 25.8T + 7.92e3T^{2} \)
97 \( 1 + (48.2 - 83.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.88465038368547759776340693703, −10.08159043918489944870553414148, −9.200493804562168228816905442838, −7.78394358430430553374834093274, −6.96843315127779173680941484975, −6.22018678636770597188758872871, −5.51092851598189189259447845587, −4.23501929811159741105813942263, −2.62032945273377487952441496879, −1.57918615951967715087402001431, 0.65167973796904829623747616459, 1.75528046857123585260037572633, 3.94507163453060378997514197424, 4.74805766146823919955312296275, 5.66499498881111209574205260136, 6.23061342126333778188295764995, 7.73167091302730197980844438070, 8.643354927183824382999635130704, 9.537571365893261613431012496338, 10.27165398807788491984409335397

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