Properties

Label 2-768-24.5-c2-0-5
Degree $2$
Conductor $768$
Sign $-0.947 + 0.321i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 2.69i)3-s + 0.640·5-s − 2.72·7-s + (−5.47 − 7.14i)9-s + 11.2·11-s − 5.25i·13-s + (−0.849 + 1.72i)15-s + 14.8i·17-s + 15.0i·19-s + (3.61 − 7.31i)21-s + 36.4i·23-s − 24.5·25-s + (26.4 − 5.24i)27-s − 51.7·29-s − 36.5·31-s + ⋯
L(s)  = 1  + (−0.442 + 0.896i)3-s + 0.128·5-s − 0.388·7-s + (−0.608 − 0.793i)9-s + 1.02·11-s − 0.403i·13-s + (−0.0566 + 0.114i)15-s + 0.874i·17-s + 0.793i·19-s + (0.171 − 0.348i)21-s + 1.58i·23-s − 0.983·25-s + (0.980 − 0.194i)27-s − 1.78·29-s − 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.947 + 0.321i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.947 + 0.321i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3829371946\)
\(L(\frac12)\) \(\approx\) \(0.3829371946\)
\(L(2)\) 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 \( 1 + (1.32 - 2.69i)T \)
good5 \( 1 - 0.640T + 25T^{2} \)
7 \( 1 + 2.72T + 49T^{2} \)
11 \( 1 - 11.2T + 121T^{2} \)
13 \( 1 + 5.25iT - 169T^{2} \)
17 \( 1 - 14.8iT - 289T^{2} \)
19 \( 1 - 15.0iT - 361T^{2} \)
23 \( 1 - 36.4iT - 529T^{2} \)
29 \( 1 + 51.7T + 841T^{2} \)
31 \( 1 + 36.5T + 961T^{2} \)
37 \( 1 + 63.6iT - 1.36e3T^{2} \)
41 \( 1 - 12.1iT - 1.68e3T^{2} \)
43 \( 1 - 11.8iT - 1.84e3T^{2} \)
47 \( 1 + 61.1iT - 2.20e3T^{2} \)
53 \( 1 - 59.1T + 2.80e3T^{2} \)
59 \( 1 + 37.2T + 3.48e3T^{2} \)
61 \( 1 + 58.1iT - 3.72e3T^{2} \)
67 \( 1 - 23.0iT - 4.48e3T^{2} \)
71 \( 1 - 7.29iT - 5.04e3T^{2} \)
73 \( 1 + 73.4T + 5.32e3T^{2} \)
79 \( 1 - 58.5T + 6.24e3T^{2} \)
83 \( 1 - 32.3T + 6.88e3T^{2} \)
89 \( 1 - 112. iT - 7.92e3T^{2} \)
97 \( 1 + 80.0T + 9.40e3T^{2} \)
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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

−10.60913680329562141041592821567, −9.508762587805569940019223881422, −9.369442824915196980013391324974, −8.103923989461196413342915371766, −7.03381328852467421594638738728, −5.84272802890933791604868101417, −5.52834833132113456195973392081, −3.84491182604627162163529663848, −3.67116785639239031100910084271, −1.72353326638303979384669443132, 0.13683541060027609475154013236, 1.56768069429260979583936373307, 2.75594832343645659814325393035, 4.16833947625707017763785942081, 5.31444821122406560572198580566, 6.32436112954085145956443445125, 6.89260472711939846289962707224, 7.73725117891044975002353825401, 8.894255486293709341690720887403, 9.488245454079502051716955645014

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