Properties

Label 2-768-24.11-c3-0-86
Degree $2$
Conductor $768$
Sign $-0.902 + 0.430i$
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

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.902 + 0.430i$
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: \((2,\ 768,\ (\ :3/2),\ -0.902 + 0.430i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.170399490\)
\(L(\frac12)\) \(\approx\) \(2.170399490\)
\(L(\frac{5}{2})\) 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

−9.667368956916797440254806451830, −8.707877972469686311367590951963, −7.70894331753783085043801800313, −6.91718907421720168809358457676, −6.19066131393671017445442640452, −5.27775505244812536297838920783, −3.94442633369084435905572428154, −2.45729519506395321709430386301, −1.81970400463637963625862868593, −0.49367028645149247083518213242, 1.85614044847558988350764716628, 2.58179947070366455719163306291, 3.87953960702055116451319943245, 4.98452663086178441223796523749, 6.08972459595890404212151665181, 6.16992779629884449713556985055, 8.199425457740540967372567701581, 8.695581312322598988026997958824, 9.515173039482341453427869395633, 10.13097630215496626157827270271

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