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Properties

Label 2-768-12.11-c3-0-10
Degree $2$
Conductor $768$
Sign $-0.853 + 0.520i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.70 + 4.43i)3-s + 19.2i·5-s − 13.3i·7-s + (−12.3 + 24i)9-s + 22.3·11-s − 39.3·13-s + (−85.4 + 52.1i)15-s + 122. i·17-s − 107. i·19-s + (59.3 − 36.1i)21-s − 21.0·23-s − 246.·25-s + (−139. + 10.0i)27-s − 72.0i·29-s + 203. i·31-s + ⋯
L(s)  = 1  + (0.520 + 0.853i)3-s + 1.72i·5-s − 0.721i·7-s + (−0.458 + 0.888i)9-s + 0.611·11-s − 0.840·13-s + (−1.47 + 0.896i)15-s + 1.75i·17-s − 1.29i·19-s + (0.616 − 0.375i)21-s − 0.190·23-s − 1.96·25-s + (−0.997 + 0.0715i)27-s − 0.461i·29-s + 1.17i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.853 + 0.520i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.853 + 0.520i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.274115821\)
\(L(\frac12)\) \(\approx\) \(1.274115821\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.70 - 4.43i)T \)
good5 \( 1 - 19.2iT - 125T^{2} \)
7 \( 1 + 13.3iT - 343T^{2} \)
11 \( 1 - 22.3T + 1.33e3T^{2} \)
13 \( 1 + 39.3T + 2.19e3T^{2} \)
17 \( 1 - 122. iT - 4.91e3T^{2} \)
19 \( 1 + 107. iT - 6.85e3T^{2} \)
23 \( 1 + 21.0T + 1.21e4T^{2} \)
29 \( 1 + 72.0iT - 2.43e4T^{2} \)
31 \( 1 - 203. iT - 2.97e4T^{2} \)
37 \( 1 + 205.T + 5.06e4T^{2} \)
41 \( 1 - 21.0iT - 6.89e4T^{2} \)
43 \( 1 - 276. iT - 7.95e4T^{2} \)
47 \( 1 + 533.T + 1.03e5T^{2} \)
53 \( 1 + 240. iT - 1.48e5T^{2} \)
59 \( 1 - 478.T + 2.05e5T^{2} \)
61 \( 1 - 33.5T + 2.26e5T^{2} \)
67 \( 1 + 261. iT - 3.00e5T^{2} \)
71 \( 1 + 470.T + 3.57e5T^{2} \)
73 \( 1 - 354.T + 3.89e5T^{2} \)
79 \( 1 + 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 - 590.T + 5.71e5T^{2} \)
89 \( 1 + 516. iT - 7.04e5T^{2} \)
97 \( 1 + 272.T + 9.12e5T^{2} \)
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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

−10.34866622815869372043815372904, −9.889388023618913601007031879420, −8.795266515664620388505524297608, −7.79664314275324632274251426714, −6.96614685577048028057743048628, −6.24902542873416490136591252757, −4.82657224774668778311378439848, −3.80250394167636450097863225314, −3.13130839589312564823310290292, −2.02306772755888604306049923644, 0.30608085676131999615392104893, 1.42684433117986969351209002218, 2.43252961202509298674663486933, 3.85221034254891611494697653630, 5.05835793573533338822792857902, 5.72711575666500265217209425365, 6.95153554097923998933874337205, 7.87640740991658431103306214674, 8.591562103259632449698767021006, 9.281107999675921271205332110488

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