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Properties

Label 2-768-12.11-c1-0-21
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 + 1.41i)3-s − 2.82i·5-s − 2.82i·7-s + (−1.00 + 2.82i)9-s + 2·11-s − 4·13-s + (4.00 − 2.82i)15-s − 5.65i·17-s − 2.82i·19-s + (4.00 − 2.82i)21-s + 8·23-s − 3.00·25-s + (−5.00 + 1.41i)27-s + 2.82i·29-s − 8.48i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s − 1.26i·5-s − 1.06i·7-s + (−0.333 + 0.942i)9-s + 0.603·11-s − 1.10·13-s + (1.03 − 0.730i)15-s − 1.37i·17-s − 0.648i·19-s + (0.872 − 0.617i)21-s + 1.66·23-s − 0.600·25-s + (−0.962 + 0.272i)27-s + 0.525i·29-s − 1.52i·31-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

\(L(1)\) \(\approx\) \(1.51855 - 0.786063i\)
\(L(\frac12)\) \(\approx\) \(1.51855 - 0.786063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
good5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 5.65iT - 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2.82iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.48iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.953089882344299536716434294638, −9.263378320586195366362382753612, −8.819446980705997261546477777318, −7.62001859279758778663206024957, −7.00130961375351693650600377747, −5.21517854425873512394274344406, −4.76938340074289338914595762334, −3.89946518316382175569935990499, −2.61303592544926057220157268307, −0.835778173840597778927808714701, 1.79918803859608130451973173990, 2.77733901584989806329493698843, 3.58652627780453022808913128848, 5.26960162720519770499206824585, 6.42927177558659482632124989552, 6.83763238364471673847100831033, 7.84788323485869656032738681067, 8.686909875576544106754432464652, 9.476957162199833906927255029521, 10.45943174528697459527542872978

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