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Properties

Label 2-768-12.11-c3-0-12
Degree $2$
Conductor $768$
Sign $-0.881 + 0.471i$
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  + (−4.58 + 2.44i)3-s + 17.2i·5-s + 0.269i·7-s + (15 − 22.4i)9-s − 41.1·11-s + 64.6·13-s + (−42.3 − 79.1i)15-s + 63.4i·17-s + 154. i·19-s + (−0.660 − 1.23i)21-s + 43.3·23-s − 173.·25-s + (−13.7 + 139. i)27-s + 240. i·29-s + 79.1i·31-s + ⋯
L(s)  = 1  + (−0.881 + 0.471i)3-s + 1.54i·5-s + 0.0145i·7-s + (0.555 − 0.831i)9-s − 1.12·11-s + 1.37·13-s + (−0.728 − 1.36i)15-s + 0.904i·17-s + 1.85i·19-s + (−0.00686 − 0.0128i)21-s + 0.392·23-s − 1.38·25-s + (−0.0979 + 0.995i)27-s + 1.54i·29-s + 0.458i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.881 + 0.471i$
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: \((2,\ 768,\ (\ :3/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9024653205\)
\(L(\frac12)\) \(\approx\) \(0.9024653205\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.58 - 2.44i)T \)
good5 \( 1 - 17.2iT - 125T^{2} \)
7 \( 1 - 0.269iT - 343T^{2} \)
11 \( 1 + 41.1T + 1.33e3T^{2} \)
13 \( 1 - 64.6T + 2.19e3T^{2} \)
17 \( 1 - 63.4iT - 4.91e3T^{2} \)
19 \( 1 - 154. iT - 6.85e3T^{2} \)
23 \( 1 - 43.3T + 1.21e4T^{2} \)
29 \( 1 - 240. iT - 2.43e4T^{2} \)
31 \( 1 - 79.1iT - 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 - 101. iT - 6.89e4T^{2} \)
43 \( 1 + 120. iT - 7.95e4T^{2} \)
47 \( 1 + 293.T + 1.03e5T^{2} \)
53 \( 1 + 6.17iT - 1.48e5T^{2} \)
59 \( 1 - 45.8T + 2.05e5T^{2} \)
61 \( 1 - 651.T + 2.26e5T^{2} \)
67 \( 1 + 685. iT - 3.00e5T^{2} \)
71 \( 1 + 836.T + 3.57e5T^{2} \)
73 \( 1 + 285.T + 3.89e5T^{2} \)
79 \( 1 + 940. iT - 4.93e5T^{2} \)
83 \( 1 - 1.01e3T + 5.71e5T^{2} \)
89 \( 1 - 432. iT - 7.04e5T^{2} \)
97 \( 1 - 1.00e3T + 9.12e5T^{2} \)
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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

−10.69828703453079549852821169076, −10.05453673594717772884298102285, −8.711051197657415746803412918150, −7.71872892004024513719653502585, −6.72055966304397089878228039633, −6.06433571395208365687247327186, −5.28981119770005276414760110408, −3.80925241086886394325228424841, −3.22319864545178764132126440414, −1.58457680004004159796738553740, 0.33135547669272079597027127741, 1.03292692056732123486381868924, 2.46343776027986272269691549800, 4.26295030951424834453725778204, 5.07233150486236461373443857088, 5.64709615657516201014301529439, 6.75876297024941521704589849012, 7.75840731378432699950997908396, 8.549785041201801321315257566899, 9.333567989405588281408937596805

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