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Properties

Label 2-768-12.11-c3-0-57
Degree $2$
Conductor $768$
Sign $0.962 - 0.272i$
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  + (1.41 + 5i)3-s + (−23 + 14.1i)9-s + 70.7·11-s − 107. i·17-s − 106i·19-s + 125·25-s + (−103. − 95i)27-s + (100. + 353. i)33-s + 56.5i·41-s − 290i·43-s + 343·49-s + (537. − 152i)51-s + (530 − 149. i)57-s + 325.·59-s − 70i·67-s + ⋯
L(s)  = 1  + (0.272 + 0.962i)3-s + (−0.851 + 0.523i)9-s + 1.93·11-s − 1.53i·17-s − 1.27i·19-s + 25-s + (−0.735 − 0.677i)27-s + (0.527 + 1.86i)33-s + 0.215i·41-s − 1.02i·43-s + 49-s + (1.47 − 0.417i)51-s + (1.23 − 0.348i)57-s + 0.717·59-s − 0.127i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.962 - 0.272i$
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: \((2,\ 768,\ (\ :3/2),\ 0.962 - 0.272i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.452047054\)
\(L(\frac12)\) \(\approx\) \(2.452047054\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.41 - 5i)T \)
good5 \( 1 - 125T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 - 70.7T + 1.33e3T^{2} \)
13 \( 1 + 2.19e3T^{2} \)
17 \( 1 + 107. iT - 4.91e3T^{2} \)
19 \( 1 + 106iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + 5.06e4T^{2} \)
41 \( 1 - 56.5iT - 6.89e4T^{2} \)
43 \( 1 + 290iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 - 325.T + 2.05e5T^{2} \)
61 \( 1 + 2.26e5T^{2} \)
67 \( 1 + 70iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 430T + 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 681.T + 5.71e5T^{2} \)
89 \( 1 - 1.32e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.91e3T + 9.12e5T^{2} \)
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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

−9.753320271969363127831323095174, −9.126888886650275190763349867988, −8.683901965761052920588324098434, −7.26991146253385528386138527318, −6.53846826699607259004989635913, −5.26807208076078694667529097722, −4.47386191744277633701306592946, −3.54707222972038038046170602756, −2.49267657040859462314327459796, −0.78578255270400873106345616900, 1.08671366896677629974371244279, 1.85664890348767995639481792744, 3.36555950349182472985105137782, 4.18164382761516558379980819551, 5.82523191190479630978365329970, 6.40811559928550661535380054751, 7.21521096447116099154626349203, 8.285176604742706739841385493159, 8.814414519579568651895613287503, 9.763479889289445583355955255078

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