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Properties

Label 2-768-16.3-c2-0-13
Degree $2$
Conductor $768$
Sign $0.991 - 0.130i$
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  + (−1.22 + 1.22i)3-s + (2 − 2i)5-s + 0.378·7-s − 2.99i·9-s + (10.5 + 10.5i)11-s + (−6.46 − 6.46i)13-s + 4.89i·15-s − 11.8·17-s + (18.6 − 18.6i)19-s + (−0.464 + 0.464i)21-s + 12.0·23-s + 17i·25-s + (3.67 + 3.67i)27-s + (−1.07 − 1.07i)29-s + 6.38i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.400 − 0.400i)5-s + 0.0541·7-s − 0.333i·9-s + (0.959 + 0.959i)11-s + (−0.497 − 0.497i)13-s + 0.326i·15-s − 0.697·17-s + (0.982 − 0.982i)19-s + (−0.0221 + 0.0221i)21-s + 0.524·23-s + 0.680i·25-s + (0.136 + 0.136i)27-s + (−0.0369 − 0.0369i)29-s + 0.206i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.797378099\)
\(L(\frac12)\) \(\approx\) \(1.797378099\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T \)
good5 \( 1 + (-2 + 2i)T - 25iT^{2} \)
7 \( 1 - 0.378T + 49T^{2} \)
11 \( 1 + (-10.5 - 10.5i)T + 121iT^{2} \)
13 \( 1 + (6.46 + 6.46i)T + 169iT^{2} \)
17 \( 1 + 11.8T + 289T^{2} \)
19 \( 1 + (-18.6 + 18.6i)T - 361iT^{2} \)
23 \( 1 - 12.0T + 529T^{2} \)
29 \( 1 + (1.07 + 1.07i)T + 841iT^{2} \)
31 \( 1 - 6.38iT - 961T^{2} \)
37 \( 1 + (-35.3 + 35.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 3.56iT - 1.68e3T^{2} \)
43 \( 1 + (-29.9 - 29.9i)T + 1.84e3iT^{2} \)
47 \( 1 + 31.6iT - 2.20e3T^{2} \)
53 \( 1 + (-68.7 + 68.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (-27.8 - 27.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (-62.8 - 62.8i)T + 3.72e3iT^{2} \)
67 \( 1 + (-30.5 + 30.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 131.T + 5.04e3T^{2} \)
73 \( 1 - 50.1iT - 5.32e3T^{2} \)
79 \( 1 - 114. iT - 6.24e3T^{2} \)
83 \( 1 + (-1.92 + 1.92i)T - 6.88e3iT^{2} \)
89 \( 1 - 74iT - 7.92e3T^{2} \)
97 \( 1 + 89.2T + 9.40e3T^{2} \)
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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

−9.881534777467012043613576174451, −9.486702481930951273227857803113, −8.682103338970216034451572994990, −7.34719636274574118392715147426, −6.70875814498302141087866265241, −5.47602136237517958308114220868, −4.84484266996767775008252946904, −3.82673910048615659280160374143, −2.37611441884771502520785211133, −0.917990490622027645368574507379, 0.952337058999402025176999280486, 2.27154344487644551892242111545, 3.54622248218493193823939605606, 4.74990777832657542592598939332, 5.92574430824557298050303226953, 6.47280634081585860485199534069, 7.37744549944068848038313639822, 8.385016555950530844359591877675, 9.318513650450400992463073363386, 10.09943618571223850657148753189

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