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Properties

Label 2-768-12.11-c1-0-25
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

\(L(1)\) \(\approx\) \(0.853101 - 1.64806i\)
\(L(\frac12)\) \(\approx\) \(0.853101 - 1.64806i\)
\(L(\frac{3}{2})\) 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

−9.855412101229841779083687311730, −8.994596815513396479366892225998, −8.182631434351678903266495681889, −7.76900236477270828771675802500, −6.44171877092891243843215888464, −5.84569945370550628814190632473, −4.15801066384631971584738330440, −3.77015201184958142725796690470, −1.80494644618733160370409467942, −0.944281193568862877142394867146, 2.25943473043660764430411741962, 3.07660828285776428562006821508, 4.00130827318759063374247396910, 5.27042862733379677804268421892, 6.22670110820432856620595626044, 7.12465929549730367019309647976, 8.281710285438635550797649542370, 8.961369816676942957874767219788, 9.765423492759884298144469063424, 10.51790162432362088151821340791

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