Properties

Label 2-768-32.5-c1-0-14
Degree $2$
Conductor $768$
Sign $-0.932 + 0.360i$
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  + (−0.382 − 0.923i)3-s + (−2.14 − 0.890i)5-s + (1.10 + 1.10i)7-s + (−0.707 + 0.707i)9-s + (0.999 − 2.41i)11-s + (2.03 − 0.841i)13-s + 2.32i·15-s − 5.68i·17-s + (−6.02 + 2.49i)19-s + (0.595 − 1.43i)21-s + (−3.60 + 3.60i)23-s + (0.293 + 0.293i)25-s + (0.923 + 0.382i)27-s + (−3.82 − 9.23i)29-s − 1.98·31-s + ⋯
L(s)  = 1  + (−0.220 − 0.533i)3-s + (−0.961 − 0.398i)5-s + (0.415 + 0.415i)7-s + (−0.235 + 0.235i)9-s + (0.301 − 0.727i)11-s + (0.563 − 0.233i)13-s + 0.600i·15-s − 1.37i·17-s + (−1.38 + 0.572i)19-s + (0.129 − 0.313i)21-s + (−0.751 + 0.751i)23-s + (0.0586 + 0.0586i)25-s + (0.177 + 0.0736i)27-s + (−0.710 − 1.71i)29-s − 0.357·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.932 + 0.360i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111622 - 0.598459i\)
\(L(\frac12)\) \(\approx\) \(0.111622 - 0.598459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.382 + 0.923i)T \)
good5 \( 1 + (2.14 + 0.890i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.10 - 1.10i)T + 7iT^{2} \)
11 \( 1 + (-0.999 + 2.41i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.03 + 0.841i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.68iT - 17T^{2} \)
19 \( 1 + (6.02 - 2.49i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.60 - 3.60i)T - 23iT^{2} \)
29 \( 1 + (3.82 + 9.23i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.98T + 31T^{2} \)
37 \( 1 + (5.97 + 2.47i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.33 - 4.33i)T - 41iT^{2} \)
43 \( 1 + (4.39 - 10.6i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.32iT - 47T^{2} \)
53 \( 1 + (0.802 - 1.93i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.97 - 2.47i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.53 - 8.54i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.25 + 5.44i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (2.57 + 2.57i)T + 71iT^{2} \)
73 \( 1 + (-8.01 + 8.01i)T - 73iT^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (-2.50 + 1.03i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.98 + 2.98i)T + 89iT^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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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

−9.952667279100527028828570545306, −8.803446928135186264276805946283, −8.196251951052777597043899200614, −7.54939532416592665724141681366, −6.36895344344129727646355635616, −5.57437819532354249520990478935, −4.43718190740988813060498702207, −3.44100735458114815574759396721, −1.92258889978051956549858659000, −0.31044414777762346064858649128, 1.86820021173226021460848399412, 3.69770460234681381608651815272, 4.07657933718153633383932936139, 5.18072514689969536768508846050, 6.49728190400802619963101844504, 7.13605319831255152327307938672, 8.294824574919158127966069102028, 8.803880376590883658398977100766, 10.11454817509818396545234134566, 10.78617251687994480517495482929

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