↑

Properties

Label 2-768-64.53-c1-0-6
Degree $2$
Conductor $768$
Sign $0.960 - 0.278i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.980 + 0.195i)3-s + (1.62 + 2.43i)5-s + (−0.294 − 0.121i)7-s + (0.923 − 0.382i)9-s + (1.07 − 5.42i)11-s + (2.67 − 4.00i)13-s + (−2.06 − 2.06i)15-s + (0.394 − 0.394i)17-s + (2.13 + 1.42i)19-s + (0.312 + 0.0621i)21-s + (3.37 + 8.14i)23-s + (−1.36 + 3.28i)25-s + (−0.831 + 0.555i)27-s + (0.411 + 2.06i)29-s + 1.31i·31-s + ⋯
L(s)  = 1  + (−0.566 + 0.112i)3-s + (0.726 + 1.08i)5-s + (−0.111 − 0.0460i)7-s + (0.307 − 0.127i)9-s + (0.325 − 1.63i)11-s + (0.742 − 1.11i)13-s + (−0.534 − 0.534i)15-s + (0.0956 − 0.0956i)17-s + (0.490 + 0.327i)19-s + (0.0681 + 0.0135i)21-s + (0.703 + 1.69i)23-s + (−0.272 + 0.656i)25-s + (−0.160 + 0.106i)27-s + (0.0764 + 0.384i)29-s + 0.235i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.960 - 0.278i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.960 - 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51986 + 0.215885i\)
\(L(\frac12)\) \(\approx\) \(1.51986 + 0.215885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.980 - 0.195i)T \)
good5 \( 1 + (-1.62 - 2.43i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (0.294 + 0.121i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.07 + 5.42i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-2.67 + 4.00i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 + (-0.394 + 0.394i)T - 17iT^{2} \)
19 \( 1 + (-2.13 - 1.42i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-3.37 - 8.14i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.411 - 2.06i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 1.31iT - 31T^{2} \)
37 \( 1 + (-1.47 + 0.985i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-4.39 - 10.6i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.25 - 0.448i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-5.23 + 5.23i)T - 47iT^{2} \)
53 \( 1 + (-1.88 + 9.48i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (-0.373 - 0.559i)T + (-22.5 + 54.5i)T^{2} \)
61 \( 1 + (-0.868 + 0.172i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (10.3 - 2.04i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (-12.4 - 5.14i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (14.5 - 6.01i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.55 - 2.55i)T + 79iT^{2} \)
83 \( 1 + (0.585 + 0.391i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (-3.72 + 8.98i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 0.565iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43332295144939095523441433583, −9.744497436705660758148795581768, −8.715598653963645165447715304803, −7.68285169721166849911502949497, −6.65904750895248125410107540657, −5.88161349798880306814744209910, −5.39934014217741040575243156721, −3.58665769955799289174516411294, −2.99046955430541156236499659562, −1.13577559450271191067490541706, 1.16781411142978028441310174321, 2.25143138183338160108632395596, 4.28076974359735566674600148275, 4.76018854478933333698313707332, 5.88014000455110810651842098439, 6.67167247389949090746369775834, 7.57256988239963467702368485641, 9.002326476003727237438261889212, 9.218175371425655757017791524502, 10.23296240579239498048086348334

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