Properties

Label 2-768-8.3-c2-0-10
Degree $2$
Conductor $768$
Sign $0.707 - 0.707i$
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  + 1.73·3-s − 1.36i·5-s + 1.24i·7-s + 2.99·9-s − 5.79·11-s + 16.3i·13-s − 2.36i·15-s − 5.01·17-s + 26.1·19-s + 2.15i·21-s + 25.1i·23-s + 23.1·25-s + 5.19·27-s + 32.7i·29-s + 1.01i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.272i·5-s + 0.177i·7-s + 0.333·9-s − 0.527·11-s + 1.26i·13-s − 0.157i·15-s − 0.294·17-s + 1.37·19-s + 0.102i·21-s + 1.09i·23-s + 0.925·25-s + 0.192·27-s + 1.13i·29-s + 0.0328i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.178223586\)
\(L(\frac12)\) \(\approx\) \(2.178223586\)
\(L(2)\) 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 \( 1 - 1.73T \)
good5 \( 1 + 1.36iT - 25T^{2} \)
7 \( 1 - 1.24iT - 49T^{2} \)
11 \( 1 + 5.79T + 121T^{2} \)
13 \( 1 - 16.3iT - 169T^{2} \)
17 \( 1 + 5.01T + 289T^{2} \)
19 \( 1 - 26.1T + 361T^{2} \)
23 \( 1 - 25.1iT - 529T^{2} \)
29 \( 1 - 32.7iT - 841T^{2} \)
31 \( 1 - 1.01iT - 961T^{2} \)
37 \( 1 + 14.9iT - 1.36e3T^{2} \)
41 \( 1 - 72.5T + 1.68e3T^{2} \)
43 \( 1 + 33.4T + 1.84e3T^{2} \)
47 \( 1 + 66.5iT - 2.20e3T^{2} \)
53 \( 1 - 54.6iT - 2.80e3T^{2} \)
59 \( 1 + 20.5T + 3.48e3T^{2} \)
61 \( 1 - 111. iT - 3.72e3T^{2} \)
67 \( 1 - 60.9T + 4.48e3T^{2} \)
71 \( 1 + 80.4iT - 5.04e3T^{2} \)
73 \( 1 + 30.0T + 5.32e3T^{2} \)
79 \( 1 - 80.9iT - 6.24e3T^{2} \)
83 \( 1 + 113.T + 6.88e3T^{2} \)
89 \( 1 - 21.0T + 7.92e3T^{2} \)
97 \( 1 - 160.T + 9.40e3T^{2} \)
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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

−10.08157658884069369034058818169, −9.168532462902816005065965144918, −8.774713638300959652997987225723, −7.55666137618480044871614741872, −7.00713020202512827178094445883, −5.69182130147791028771608919322, −4.78984255550321175362242367867, −3.69239314990055361126645869615, −2.58500709697493714937025706959, −1.31720028970895984879773677167, 0.77277943973742958823157742571, 2.51153292759672768766481950477, 3.25041601879863790242149026798, 4.51741853919127821831025142002, 5.52556637010224764537660065908, 6.60650851534584841660543298489, 7.64662412910167011140008984624, 8.132625848002627372413619995231, 9.195364571922283933530052168947, 10.05345094225084982132001179871

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