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Properties

Label 2-768-12.11-c1-0-2
Degree $2$
Conductor $768$
Sign $0.577 - 0.816i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
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 − i)3-s + (1.00 + 2.82i)9-s − 2.82·11-s + 5.65i·17-s + 2i·19-s + 5·25-s + (1.41 − 5.00i)27-s + (4.00 + 2.82i)33-s + 11.3i·41-s + 10i·43-s + 7·49-s + (5.65 − 8.00i)51-s + (2 − 2.82i)57-s + 14.1·59-s + 14i·67-s + ⋯
L(s)  = 1  + (−0.816 − 0.577i)3-s + (0.333 + 0.942i)9-s − 0.852·11-s + 1.37i·17-s + 0.458i·19-s + 25-s + (0.272 − 0.962i)27-s + (0.696 + 0.492i)33-s + 1.76i·41-s + 1.52i·43-s + 49-s + (0.792 − 1.12i)51-s + (0.264 − 0.374i)57-s + 1.84·59-s + 1.71i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773537 + 0.400412i\)
\(L(\frac12)\) \(\approx\) \(0.773537 + 0.400412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.55179000133692586262514575894, −9.873826799037773687851782066482, −8.476828726797859763885411463056, −7.88179015385990677364770329742, −6.87910845864649643757293458251, −6.05071528493162728347776000837, −5.24882506383316024809690372661, −4.20664202398672353855469575432, −2.68191299890414399250769527500, −1.31184660084673937374665267419, 0.53428567744807779713219705668, 2.57633293308453698005659839446, 3.82977912038039599091042869748, 5.01467525339356990467809147284, 5.44908602401888469548068597467, 6.72970846236149838163561521362, 7.38401651784986402357208674302, 8.703796018131275315760272578962, 9.419750761933551320894071954498, 10.38473443645050262387954366298

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