Properties

Label 2-768-24.11-c1-0-22
Degree $2$
Conductor $768$
Sign $0.408 + 0.912i$
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  + (−0.618 + 1.61i)3-s + 1.23·5-s − 3.23i·7-s + (−2.23 − 2.00i)9-s + 0.763i·11-s − 4.47i·13-s + (−0.763 + 2.00i)15-s − 6.47i·17-s − 5.23·19-s + (5.23 + 2.00i)21-s − 6.47·23-s − 3.47·25-s + (4.61 − 2.38i)27-s + 9.23·29-s − 0.763i·31-s + ⋯
L(s)  = 1  + (−0.356 + 0.934i)3-s + 0.552·5-s − 1.22i·7-s + (−0.745 − 0.666i)9-s + 0.230i·11-s − 1.24i·13-s + (−0.197 + 0.516i)15-s − 1.56i·17-s − 1.20·19-s + (1.14 + 0.436i)21-s − 1.34·23-s − 0.694·25-s + (0.888 − 0.458i)27-s + 1.71·29-s − 0.137i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913252 - 0.591998i\)
\(L(\frac12)\) \(\approx\) \(0.913252 - 0.591998i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.61i)T \)
good5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 + 3.23iT - 7T^{2} \)
11 \( 1 - 0.763iT - 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 + 6.47iT - 17T^{2} \)
19 \( 1 + 5.23T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 - 9.23T + 29T^{2} \)
31 \( 1 + 0.763iT - 31T^{2} \)
37 \( 1 + 0.472iT - 37T^{2} \)
41 \( 1 + 2.47iT - 41T^{2} \)
43 \( 1 + 2.76T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 - 3.23iT - 59T^{2} \)
61 \( 1 - 8.47iT - 61T^{2} \)
67 \( 1 - 3.70T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 13.7iT - 79T^{2} \)
83 \( 1 + 7.23iT - 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 8.47T + 97T^{2} \)
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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

−10.26474168826005923054657301234, −9.645917022873296553987541936459, −8.555171463989951912369394409539, −7.58466529537121823514154997726, −6.53121168805146507199738683746, −5.63239728104520187500111972275, −4.66566724767796312911858904128, −3.85816401265561174507361954474, −2.60534742539296043482904941175, −0.55743542511046815730854656783, 1.77009556746866831490971604721, 2.39321899657492416575731562878, 4.12669712169994804630329275532, 5.44525734300318499721014167211, 6.25763693422057212140713946382, 6.59335924568600896717028934211, 8.206081003599359416643207846690, 8.475083803573675684595755008003, 9.577893553799522600333893612299, 10.56829192192291094139652325805

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