Properties

Label 2-768-24.11-c3-0-44
Degree $2$
Conductor $768$
Sign $-0.430 + 0.902i$
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  + (−1.73 − 4.89i)3-s − 16.9·5-s + 17.3i·7-s + (−20.9 + 16.9i)9-s + 29.3i·11-s − 26i·13-s + (29.3 + 83.1i)15-s + 67.8i·17-s + 107.·19-s + (84.8 − 30i)21-s − 176.·23-s + 162.·25-s + (119. + 73.4i)27-s − 16.9·29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (−0.333 − 0.942i)3-s − 1.51·5-s + 0.935i·7-s + (−0.777 + 0.628i)9-s + 0.805i·11-s − 0.554i·13-s + (0.505 + 1.43i)15-s + 0.968i·17-s + 1.29·19-s + (0.881 − 0.311i)21-s − 1.59·23-s + 1.30·25-s + (0.851 + 0.523i)27-s − 0.108·29-s − 0.180i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.430 + 0.902i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4479691018\)
\(L(\frac12)\) \(\approx\) \(0.4479691018\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.73 + 4.89i)T \)
good5 \( 1 + 16.9T + 125T^{2} \)
7 \( 1 - 17.3iT - 343T^{2} \)
11 \( 1 - 29.3iT - 1.33e3T^{2} \)
13 \( 1 + 26iT - 2.19e3T^{2} \)
17 \( 1 - 67.8iT - 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 176.T + 1.21e4T^{2} \)
29 \( 1 + 16.9T + 2.43e4T^{2} \)
31 \( 1 + 31.1iT - 2.97e4T^{2} \)
37 \( 1 + 206iT - 5.06e4T^{2} \)
41 \( 1 - 305. iT - 6.89e4T^{2} \)
43 \( 1 + 93.5T + 7.95e4T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 - 50.9T + 1.48e5T^{2} \)
59 \( 1 + 558. iT - 2.05e5T^{2} \)
61 \( 1 - 278iT - 2.26e5T^{2} \)
67 \( 1 + 890.T + 3.00e5T^{2} \)
71 \( 1 + 58.7T + 3.57e5T^{2} \)
73 \( 1 - 422T + 3.89e5T^{2} \)
79 \( 1 + 668. iT - 4.93e5T^{2} \)
83 \( 1 - 29.3iT - 5.71e5T^{2} \)
89 \( 1 - 373. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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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

−9.614113316353245214094108877671, −8.392982971267733534003124708451, −7.913609340397279124212099989964, −7.25428255859916150829740352249, −6.15183997337028813214386211669, −5.29882455403097179864816718849, −4.11348427231360995386539016793, −2.95662048030366176778689802953, −1.69457924893985888846738198097, −0.19035141056870628824150390883, 0.77121828348279385657970228768, 3.13515683055774566902171627809, 3.86079985609701950302077965268, 4.53287060146170668986794333082, 5.59677133943850935801405909708, 6.84927838871938849854730571535, 7.66124271177542859010893383745, 8.468246910137442773656350872886, 9.445248283749406431225810798470, 10.29767487003739323402164066058

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